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MATHEMATICAL 


DRAWING  INSTRUMENTS, 

AND 


HOW  TO  USE  THEM. 


‘^Drawing  is  the  A B C of  the  architect,  engineer,  and  surveyor." 

Sir  Isambard  Brunel. 

“Drawing  supplies  us  with  a power  whereby  long  descriptions  and 
pages  of  writing  are  at  once  superseded,  and  thus  it  is  a condensed 
shorthand  as  well  as  a universal  language.” 


B.  Bedgrave,  B.A. 


MATHEMATICAL 


DRAWING  INSTRUMENTS, 

AND 

HOW  TO  USE  THEM. 


f 


BY 


F^EDWAED  HULME,  F.L.S.,  F.S.A., 


ART-MASTER  OF  MARLBOROUGH  COLLEGE  ; 

AUTHOR  OF  “principles  OF  ORNAMENTAL  ART,”  “FAMILIAR  WILD  FLOWERS,” 
“ SUGGESTIONS  IN  FLORAL  DESIGN,”  ETC. 


SECOND  EDITION. 


APPROVED  BY 


THE  SCIENCE 


AND  ART 


DEPARTMENT. 


NEW  YORK: 

W.  T.  COMSTOCK, 

194  BROADWAY. 

1S82. 


! "f&h  3o  /VtA'  5 


74  f 


INTRODUCTION. 


The  use  of  mathematical  instruments  enters  so  largely  into 
various  kinds  of  technical  drawing,  that  some  few  suggestions 
as  to  their  employment  cannot  hut  be  of  service  to  many  who 
find  themselves  for  the  first  time  in  their  lives  the  possessors 
of  a box  of  drawing  instruments,  and  who  therefore  have  all 
their  experience  yet  to  learn.  Having  for  many  years  been 
engaged  in  teaching  the  use  of  such  things,  and  thereby 
become  acquainted  with  the  difficulties  of  the  novice,  we 
would  desire  to  give  all  such  the  benefit  of  our  own  experi- 
ence, and  to  smooth  their  path  before  them  as  far  as  may  be 
possible. 

The  student  who  provides  his  own  things  is  at  once  met  on 

\ 

the  very  threshold  by  a difficulty — the  choice  of  a suitable 
box  of  instruments.  He  sees  in  the  shop-windows  a card  of 
things  marked  ''one  shilling  the  set;”  and,  on  the  other 


701299 


INTRODUCTION, 


hand,  in  consulting  the  catalogue  of  a first-class  maker,  he 
finds  that  even  twenty  guineas  would  not  buy  some  of  the 
sets  enumerated  with  such  tempting  richness  of  detail.  Some- 
where between  these  extremes  is  thq  very  thing  he  wants, 
but  where  the  happy  mean  may  be  is  a mystery  to  him. 

It  will  be  noticed  that  we  assign  the  true  position  of  the 
hoped-for  box  somewhere  between  the  extremes;  for  we 
would  at  once  hasten  to  say  that  few  things  are  so  dear  as 
cheap  instruments.  The  legitimate  difi&culties  of  drawing 
with  instruments  are  sufidciently  great  to  the  beginner  with- 
out complicating  them  by  the  introduction  of  pens  that  will 
not  mark,  screws  that  will  not  turn,  and  all  the  other  troubles 
that  assail  any  one  rash  enough  to  buy  things  at  a price  that 
absolutely  forbids  good  workmanship.  On  the  other  hand, 
even  where  the  pecuniary  question  raises  no  bar  to  consider- 
able expenditure,  it  is  rather  a mistake  for  the  novice  to  get 
an  expensive  box ; he  had  far  better  get  one  with  fewer  in- 
struments, and  learn  thoroughly  what  can  be  done  with  those, 
before  getting  what  may  be  considered  to  some  extent  luxuries, 
and  the  prehminary  failures  will  have  been  got  through  at 
the  risk  of  damaging  instruments  of  comparatively  small  cost. 
When  the  student  has  passed  through  his  novitiate,  has 
learned  to  take  care  of  his  things,  and  has,  moreover,  learned 
the  real  nature  of  the  work  he  has  to  do,  and  what  means 


INTRODUCTION, 


vii 

will  most  effectually  do  it,  he  can  then  go  in  for  a more  com- 
plete set  of  implements. 

Tor  a beginner,  an  expenditure  of  three  or  four  pounds 
should  give  him  all  that  is  needful  to  make  a very  effective 
start : this  should  include  a board,  T square,  &c. ; and  even 
half  this  might  in  many  cases  be  found  sufficient.  The 
surest  way  of  getting  value  for  the  money  is  to  go  at  once  to 
some  good  maker ; his  charges  will  probably  seem  somewhat 
high,  but  it  must  be  remembered  that  he  got  his  reputation 
by  the  production  of  good  things,  and  that  his  name  will  be 
a sufficient  warranty.  The  novice  should  beware  of  second- 
hand cases,  as  they  are  often  considerably  worn,  while  at 
other  times  the  name  veils  a fraud : it  is  merely  an  attempt 
to  pass  off  some  worthless  things  that  have  never  had  a pre- 
vious owner  at  all.  It  is  always  safer,  too,  to  buy  a set  that 
has  the  maker’s  name  stamped  somewhere,  either  on  the  box 
or  on  some  of  the  instruments. 

It  must  be  borne  in  mind,  in  calculating  expense,  that  when 
the  draughtsman  has  once  got  a sufficient  knowledge  of  how 
to  treat  his  instruments  to  justify  him  in  getting  a good  set 
the  expense  comes  once  for  all : unlike  the  daily  bread-and- 
butter,  an  ever-recurring  charge,  a good  box  of  instruments  is 
a possession  for  life.  The  instruments  we  ourselves  use  we 
have  had  now  some  twenty  years,  and  there  is  no  reasonable 


INTRODUCTION. 


viii 

room  to  doubt  that  another  twenty  years  may  pass  over  them 
and  find  them  less  affected  by  the  ravages  of  time  than  their 
owner. 

It  is  impossible  to  define  very  exactly  the  box  that  should 
be  procured ; but  no  set  should  be  considered  sufficient  that 
does  not  contain  compasses  suitable  for  either  pen  or  pencil 
work,  a ruling-pen,  and  a scale.  In  most  boxes  two  sizes  of 
compasses  are  found,  one  suitable  for  small  and  the  other  for 
large  circles.  The  material  of  which  the  instruments  are 
made  is  another  item  adding  more  or  less  to  the  cost ; and 
even  the  nature  of  the  case — ^walnut,  mahogany,  Eussia 
leather,  and  whatever  it  may  be  — influences  the  total 
expenditure. 

Where  it  is  possible  to  avail  oneself  of  the  advice  of  an 
experienced  friend,  it  is  clearly  a great  gain  to  do  so;  or 
where  a beginner  is  going  to  pass  into  some  definite  position, 
as  the  engineer’s  office  of  some  great  railway,  or  the  training 
at  the  Eoyal  Military  College  at  Woolwich,  he  should  en- 
deavour to  ascertain  if  any  particular  set  of  instruments 
receives  a more  especial  and  official  sanction  and  approval 
than  others,  when  he  will  do  well  to  get  it.  Assuming,  how- 
ever, that  neither  of  these  solutions  of  the  problem  can  be 
rendered  available,  that  neither  the  aid  of  a friend  can  be 
invoked  nor  the  experience  of  any  special  office  utilised,  our 


INTRODUCTION, 


IX 


student  will  do  well,  as  we  have  already  said,  to  place  himself 
in  the  hands  of  a respectable  firm,  and  what  they  will  probably 
give  him  we  now  proceed  to  analyse,  instrument  by  instru- 
ment, pointing  out  its  method  of  use,  how  it  can  be  most 
effectively  employed,  together  with  any  other  little  details 
gathered  during  a long  experience,  that  may  be  helpful  to 
those  making  their  first  steps  in  a new  direction. 


CONTENTS. 


CHAPTEE  L 

PACK 

Instruments  used  in  the  production  of  straight  lines — Euling-pen 
— Its  construction — How  filled — How  selected — Advantage 
of  having  two  pens — Indelibility  of  pen-lines — Preliminary 
pencilling  essential — Eules  for  inking-in — Trials  on  waste 
paper — Importance  of  cleanliness — How  to  clean  and  set  the 
ruling-pen — Common  ink  to  be  avoided — Dotting- wheels — 

The  bordering-pen — Eoad-pen — Section-pen  . • . i 


CHAPTEE  11. 

The  T square — Directions  for  its  use — Must  never  be  cut — 

Greek  fret-pattern — Squaring-off  a drawing — Materials  used 
in  making  T squares — Useful  size  to  get — Cost — Various 
forms  of  T square — Stanley  square — Movable-headed  square 
— The  straight-edge — Useful  size  to  get — Importance  of 
seasoned  wood — Materials  employed  for  straight-edges — 
Straight-edge  used  for  dividing  paper — How  to  test  accuracy 
of  straight-edge — Needles  for  centres  or  vanishing  points — 

The  marking  of  points  . . ♦ ....  14 


CONTENTS. 


xii 


CHAPTEE  III. 

PAGE 

The  set-square — Its  form  and  nature — Useful  size  to  get — Warp- 
ing, and  how  to  remedy  it — Materials  employed — Framed 
set-squares — Cost — How  to  use  the  set-square — Section  lines 
— The  set-square  of  45° — The  set-square  of  60° — Drawing 
nuts — Geometrical  means  of  testing  the  correctness  of  the 
set-square  — Parallels  and  perpendiculars  — Batter  lines — 

Earth  slopes — Koof  pitches — Parallel  ruler — Polling  par- 
allels— Cost 26 

CHAPTEE  IV. 

The  drawing-board — Sizes  to  get — Thorough  seasoning  of  the 
wood — Trueness  of  edge — Paper  to  be  put  truly  on — Eough 
board  lor  cutting  on — Hints  on  re-pinning  paper  down — 
Overlapping  edges  of*  paper  to  be  avoided — Materials  used 
for  boards — Cost  of  boards — Cross  strengthenings  at  back — 

Both  sides  of  a board  not  to  be  in  use  at  once — Centrolinead 
— Its  nature  and  use — Various  forms  of  it — Excentrolinead 
— Lengthening  out  lines — Drawing  lines  by  a chalked  cord 
— Line  drawings — Various  kinds  of  lines  ....  38 

CHAPTEE  V 

Instruments  for  drawing  curved  lines — Necessity  of  practice  in 
freehand  drawing-— Various  sizes  of  compasses — The  bows — 

Spring  bows — Compass  joints — Fitting  pencil  to  compass — 
Management  of  pen  point — The  lengthening  bar — Loose 
joints — Compass  key — Double-jointed  compasses — Compass 
points — Large  holes  at  centres  to  be  avoided — Horn  centres 
— Their  cost — The  use  of  the  ink  compass — Circles  to  be 
drawn  before  straight  lines  joining  them — Pocket  compasses 
— Napier  compass — Pillar  compass — Beam  compass  . 


49 


CONTENTS. 


xiii 


CHAPTEE  YI. 

PAGE 

The  circle  in  mathematical  drawing — Scale  form  and  guilloche — 

Ellipse — The  oval — Approximations  to  the  ellipse  by  means 
of  arcs — Ellipse  drawn  by  means  of  string — By  means  of  a 
strip  of  paper — The  elliptic  trammel — Conchoidograph — 
Entasis  of  columns — The  spiral  line — Methods  of  drawing  it 
— Eailway  curves — Splines  and  weights — French  curves — 
Method  of  using  them — Materials  employed  for  them — Card- 
board curves 6i 

CHAPTEE  YIL 

Dividers — Directions  for  their  use — Stepping  out  a measurement 
— Great  accuracy  essential — Geometrical  methods  for  divid- 
ing a line  into  any  number  of  equal  parts — The  division  of 
a circle — Geometrical  figures  based  on  polygons — Accumula- 
tion of  error  in  setting  out  divisions — Centre  lines — Tri- 
angular compasses — How  employed — Methods  of  drawing 
an  irregular  figure — The  pricker — Copying  drawings  by 
its  aid 71 

CHAPTEE  YIIL 

Scales,  their  nature  and  construction — The  representative  frac- 
tion— Eeading  to  edge — Duodecimal  scales  in  common  use 
— Diagonal  scales,  their  construction  and  use — Decimals — 

Other  scales  found  on  protractors,  &c.  . , . . 83 

CHAPTEE  IX. 

The  Marquois  scale,  its  construction  and  examples  of  its  use — 

Cost — Natural  and  artificial  scale — Section  lines — Propor- 


XIV 


CONTENTS. 


tional  compasses — Scale  of  lines — Enlargement  or  reduction 
of  drawings — Scale  of  circles — Division  of  circles  into  equal 
parts — Scale  of  plans — The  determination  of  areas — Scale 
of  solids — Determination  of  bulks — Cost  of  proportionals — 
Wholes  and  halves — Eidograph — Pantagraph — Measurement 
of  angles — The  protractor— Division  of  the  circle  into  degrees, 
minutes,  and  seconds — Similar  and  equal  figures — Line  of 
chords 9^ 


CHAPTER  X. 

The  sector — Principle  of  its  construction — Line  of  lines — Illus- 
trative examples  of  its  use — Enlargement  or  reduction  in 
any  given  proportion — Line  of  polygons — Illustrations  of  its 
use — Description  of  polygons  or  multilateral  figures — The 
line  of  chords — Examples  of  its  use — Cost  of  the  sector — A 
box  of  mathematical  drawing  instruments  — Second-hand 
things  — School  sets — Cost  of  various  selections  of  instru- 
ments— Long  measure — Surveyor’s  measure — Square  mea- 
sure— Solid  measure — Abbreviations — Figuring  dimensions 
on  drawings io6 


CHAPTER  XL 

Paper — Best  sizes  to  get — Cost — Hand  or  machine  made  paper — 
Cartridge  paper — Sizes  made — Mounted  papers — Straining 
paper  — Causes  of  failure  — Straining  paper  on  panelled 
boards — Drawing  pins — Care  in  re-pinning  paper  down — 
Moisture  injurious  to  paper — How  to  mount  drawings — 
Mounting  boards—  Over-mounts — Under-mounts — Tracing 
paper — Papier  v4g4tal — Directions  for  tracing — Tracing  by 
leaded  paper — Tracing  by  a sheet  of  glass — Hand  paper  . 1 19 


CONTENTS. 


XV 


CHAPTER  XII. 

Pencils — Best  kinds  to  use — How  pencils  should  be  cut — Putting 
the  pencil  in  the  mouth — Knife  and  file — Pocket-pencil — 
India-rubber  — Care  in  pencilling  — Scrolling  out  lines  — 
Bottle-rubber  — Vulcanised  rubber  — Ink-eraser  — India  - 
rubber  never  to  be  held  in  hand  when  not  in  use — How  to 
cut  Indian-rubber — Knife  erasures  to  be  avoided — Stale 
bread — Indian-ink — How  to  select  it — How  to  prepare  it 
for  use — Liquid  ink — Nests  of  saucers — Common  ink  to  be 
avoided — Common  pen  and  crowquill — Printing — How  to 
space  out  lettering — Useful  alphabets — Lettering  square — 
Arbitrary  signs  in  topographical  work — All  drawings  to  be 
signed  and  dated — Stencilling — Colouring — Ox-gall — Recog- 
nised colours  for  various  materials — Prepared  liquid  colours 
— Brushes  required — Great  cleanliness  necessary — How  to 
choose  a brush — Camel-hairs — Sables — Sizes  and  prices — 
Closing  remarks,  


132 


THE 

OF  THE 
C?  !LL!J!D!S 


MATHEMATICAL  INSTRUMENTS. 


CHAPTEE  I. 

Instruments  used  in  the  production  of  straight  lines — Euling-pen-^ 
Its  construction — How  filled — How  selected — Advantage  of  having 
two  pens — Indelibility  of  pen-lines — Preliminary  pencilling  essen- 
tial—Eules  for  inking-in — Trials  on  waste  paper — Importance  of 
cleanliness — How  to  clean  and  set  the  ruling-pen — Common  ink  to 
he  avoided — Dotting- wheels — The  hordering-pen — Eoad-pen — Sec- 
tion-pen. 

I.  Instruments  may  very  conveniently,  for  the  purposes 
of  description,  be  divided  into  those  used  in  the  production 
of  straight  lines,  those  employed  in  making  various  kinds  of 
curved  lines,  and  those  more  especially  employed  for  the 
measurement  of  lines  and  angles.  The  division  here  given 
is  only  a matter  of  convenience,  an  aid  to  systematic  descrip- 
tion, and  must  not  be  pressed  home  too  rigidly,  since  clearly 
the  compasses  that  appear  in  our  second  division,  as  the 
means  of  making  curved  lines,  may  equally  well  be  inserted  in 
our  third  section  as  a means  of  measuring  lines.  In  the  same 
way  the  protractor  or  the  set-square,  useful  as  a means  of 
drawing  straight  lines,  and  therefore  within  the  scope  of  our 
first  section,  may  equally  justly  figure  in  our  third  as  means 
of  obtaining  angles ; w^hile  some  few  instruments  we  may 
probably  find  will  scarcely  fall  satisfactorily  within  the  limits 

A 


2 


MATHEMATICAL  INSTRUMENTS, 


of  any  of  the  three  classes.  The  arrangement  will,  neverthe- 
less, be  found  in  practice  a sufficiently  eligible  one. 

2.  The  instruments  used  in  the  production  of  straight  lines 
are  the  ruling-pen,  the  T square,  the  straight-edge,  and  the 
various  forms  of  set-square.  To  these,  though  it  perhaps 
scarcely  comes  rigidly  under  the  idea  conveyed  by  the  word 
instrument,  we  may  very  legitimately  add  the  drawing-board, 
since  we  can  by  its  aid,  assisted  only  by  the  T square,  draw 
any  number  of  parallel  lines  and  any  number  of  others  at 
right  angles  to  them.  In  architectural  drawing,  for  example, 
where  so  many  of  the  lines  are  either  horizontal  or  upright, 
the  board  and  the  T square  between  them  do  almost  all 
the  work. 

3.  The  ruling-pen,  when  it  is  a good  one,  is  one  of  the 
draughtsman’s  most  cherished  possessions,  for  the  difference 
in  comfort  is  something  enormous  between  the  instrument 
that  at  once  responds  to  your  wish  and  produces  without 
any  trouble  just  the  line  you  want,  and  the  thing  that  has  to 
be  drawn  up  and  down  till  in  the  humour  to  mark,  or  that 
requires  to  be  held  just  at  one  particular  angle  before  it  can 
be  induced  to  mark  at  all.  The  fault  is,  however,  more  often 
in  the  owner  than  in  the  pen,  since  the  practised  hand  can 
carefully  ''  set  ” the  pen,  as  it  is  termed,  while  the  novice  has 
the  instruments  supplied  to  him  in  good  working  condition, 
and  it  is  ordinarily  a want  of  care  in  cleaning  that  renders 
them  less  serviceable  to  him  than  they  should  have  been. 

4.  The  ruling-pen  consists  of  two  pieces  of  metal  so  joined 
together  at  one  end  that  they  can  readily,  by  means  of  a 
screw,  be  adjusted  at  the  free  ends  to  any  required  width  for 
ruling  the  lines  of  an  ordinary  drawing.  The  intermediate 
space  holds  the  necessary  ink ; too  much  should  not,  however, 
be  taken  at  a time,  or  it  will  probably  cause  a blot.  On  com- 
mencing work  the  nibs  should  be  slightly  damped;  the  ink 


THE  RULING-PEN, 


3 


will  then  run  easily  into  its  place.  To  fill  the  pen  with  ink, 
it  may  either  be  dipped  into  a slab  having  slanting  divisions 
in  which  the  ink  has  already  been  rubbed,  or  it  may  be  sup- 
plied by  means  of  a brush;  in  either  case  the  outsides  of  the 
nibs  should  be  as  little  soiled  with  it  as  possible.  In  holding 
the  pen  for  work,  the  nib  having  the  circular  screw-head  upon 
it  is  always  outwards,  and  it  is  especially  necessary  to  see  that 
the  outer  surface  of  the  other  nib  is  perfectly  free  from  ink,  as 
it  is  this  surface  that  is  pressed  lightly  against  the  T square 
in  ruling,  and  any  lack  of  care  in  this  respect  will  almost  cer- 
tainly lead  to  a long,  ragged  smear. 

5.  In  selecting  a pen,  care  should  be  taken  to  see  that  the 
nibs  are  of  the  same  length,  and  that  they  are  of  a sufficient 
thickness.  In  some  pens  the  metal  is  so  thin  that  the  nibs 
really  cut  into  the  paper,  and  a further  disadvantage  is  that 
in  using  the  very  moderate  amount  of  pressure  necessary  to 
keep  the  back  nib  against  the  ruler  the  two  surfaces  of  metal 
become  nearly  or  quite  in  contact,  and  the  line  becomes 
uneven.  It  is  a mistake  to  select  too  small  a pen  under  the 
idea  that  it  will  necessarily  enable  the  draughtsman  to  make 
finer  lines ; a pen  of  medium  size,  if  in  proper  order,  will 
work  quite  as  finely,  and  the  larger  handle  is  far  more  con- 
venient to  hold. 

6.  In  some  boxes  of  instruments  two  pens  are  supplied. 
As  the  ruling-pen  is  so  essential  in  all  kinds  of  mathematical 
drawing,  it  is  decidedly  an  advantage  to  be  doubly  armed,  for 
several  reasons.  If,  for  instance,  anything  happens  to  so 
indispensable  an  implement  that  necessitates  it  being  re-set, 
the  possession  of  a second  in  reserve  saves  valuable  time,  as 
the  work  does  not  then  come  to  a stand-still.  In  many  out- 
line drawings,  too,  of  architectural  or  mechanical  details  it  is 
customary  to  give  what  are  termed  shadow  lines.  These  are 
thicker  than  the  others,  and  if  one  pen  is  adjusted  for  the 


4 


MATHEMATICAL  INSTRUMENTS, 


thin  lines  and  the  other  for  the  thick,  it  is  a great  saving  of 
time,  and  far  more  convenient  than  either  leaving  all  one 
series  till  the  other  is  finished,  or  having  to  keep  altering  the 
adjustment.  In  perspective  diagrams,  again,  it  is  sometimes 
advisable,  in  a complicated  drawing,  to  represent  the  object 
in  black,  and  the  lines  used  in  finding  it  in  red  or  blue,  and 
here  the  possession  of  two  pens  is  again  an  advantage,  as  one 
can  be  used  for  each  colour  employed. 

7.  In  one  of  the  army  examination  papers  we  find  a ques- 
tion that  is  almost  entirely  intended  to  test  the  use  of  the 
ruling-pen.  The  question  runs  as  follows  : — Draw  a square 
having  sides  half  an  inch  long ; ink  this  in  with  fine  lines,  and 
about  it  place  three  other  squares  parallel  to  it,  and  one-third 
of  an  inch  apart ; ink  these  in  so  that  each  square  shall  have 
a thicker  line  than  the  one  within  it.  Beginners  will  find  it 
good  practice  to  work  this  out  for  themselves.  Care  must  be 
taken  that  the  transition  from  the  inner  thin  square  to  the 
outer  thick  one  is  gradual  and  progressive ; there  must  not 
be  too  sudden  an  increase  in  any  one  square. 

8.  A very  similar  question  to  this,  and  from  the  same 
source,  is  the  following : — Draw  two  parallel  lines  one  and  a 
half  inches  apart,  divide  the  intervening  space  into  ten  equal 
parts,  and  draw  other  lines  parallel  to  the  bounding  lines, 
inking  them  in  with  thick,  thin,  and  dotted  lines  alternately. 
Like  the  preceding  figure,  we  commend  this  to  the  novice. 

9.  In  ruling  ink  lines  care  must  be  exercised  in  several 
directions.  It  must  be  borne  in  mind  by  the  beginner  that 
a line  once  drawn  in  ink  is  practicably  indelible.  Scratch- 
ing out  may  in  some  cases  be  resorted  to  where  the  error  is 
confined  to  a small  and  isolated  portion  of  the  work,  but  the 
eye  readily  detects  the  difference  of  appearance  if  the  knife 
has  been  much  in  use,  and  it  is  almost  impossible  either  to 
ink  or  colour  over  the  place  where  the  erasure  has  been  made 


CARE  IN  INKING-IN  DRA  WINGS. 


5 


without  betraying  the  fact.  Beginners  are  often  tempted  to 
hurry  on  to  the  inking ; a certain  amount  of  pencil  outlining 
has  been  done,  and  it  seems  so  easy  then  just  to  mark  off  the 
right  points,  and  then  rule  many  of  the  details  in  at  once  in 
ink.  Too  often,  however,  the  fatal  moment  arrives  ; a line  is 
taken  beyond  its  proper  termination,  a detail  of  construction 
that  should  have  gone  beneath  another  is  taken  over  it  or 
through  it ; a measurement  wrongly  taken,  a little  piece  not 
clearly  understood,  or  a moment’s  abstraction  of  the  thoughts 
from  the  work  in  hand  have  sufficed  to  work  all  the  mischief. 
The  student  begins  to  realise  the  meaning  of  the  old  saying, 
that  the  longest  way  round  may  sometimes  be  the  nearest 
way  home,  and  he  begins  his  drawing  again  full  of  good 
resolutions  to  get  it  all  well  pencilled-in  first  before  he  thinks 
of  the  inking.  How  far  he  maintains  these  good  resolutions 
will  probably  .decide  whether  the  work  this  second  time 
shall  reach  a satisfactory  termination  or  not. 

10.  To  those  of  more  experience  this  elaborate  pencilling- 
in  of  all  the  details  is  not  so  absolutely  necessary.  Where, 
as  in  the  sides  of  the  teeth  of  a cog-wheel,  a number  of 
similar  lines  all  have  to  be  drawn  to  one  point,  much  time 
may  often  be  saved  by  at  once  drawing  these  in  in  ink,  when 
their  starting-points  and  terminations  are  clearly  defined; 
but  even  to  those  of  riper  experience  misfortunes  happen, 
and  in  the  long-run  caution  pays  best. 

Beginners  often  tire  of  a long  preliminary  spell  of 'pencil 
work,  and  want  to  at  least  ink-in  what  may  be  clearly  seen 
to  be  correct.  This,  though  it  may  certainly  vary  the  mono- 
tony of  the  work  and  seem  to  give  a greater  show  of  progress, 
is  not  by  any  means  advisable,  and  the  only  time  when  it 
is  distinctly  well  to  do  it  is  when  in  a large  and  compli- 
cated drawing  the  pencil  lines  begin  to  rub  a little.  Where 
care,  however,  is  exercised,  and  the  proper  amount  of  clean- 


6 


MATHEMATICAL  INSTRUMENTS, 


liness  in  the  use  of  hand-paper,  &c.,  is  resorted  to,  there 
need  he  no  fear  of  the  blurring  and  obliteration  of  the  black- 
lead  lines. 

11.  In  inking-in,  all  objects  that  come  in  front  of  others 
and  hide  portions  of  them,  should  be  put  in  first,  and  great 
care  must  be  exercised  in  the  preliminary  sketch  in  pencil 
that  all  back  objects  should  be  clearly  marked  as  such.  It 
is  often  exceedingly  advisable  to  carry  on  a line  behind  an 
object  so  as  to  make  sure  of  the  proper  continuity  of  the 
parts  when  it  is  seen  again;  but  such  lines  should  either  be 
drawn  much  more  lightly,  made  of  a different  character,  or 
else  carefully  removed  before  the  inking  begins. 

12.  Long  lines  will  present  more  difficulty  often  to  the 
beginner  than  short  ones ; he  feels  no  hesitation  in  drawing  a 
line  in  ink  two  inches  long,  but  a line  two  feet  long  is  much 
more  formidable.  Assuming  that  the  square  or  ruler  em- 
ployed is,  as  it  should  be,  perfectly  true,  the  line  may,  never- 
theless, when  drawn  by  its  aid,  be  evidently  irregular ; this 
arises  from  a want  of  care  in  keeping  the  nibs  of  the  pen 
always  at  the  same  distance  from  the  edge  of  the  ruler. 
This,  when  stated,  appears  so  evident  a truism  as  to  be  scarcely 
worth  formally  laying  down ; it  is  nevertheless  a point  that 
in  practice  has  to  be  borne  in  mind.  A certain  portion  of 
the  back  nib  of  the  pen  will  be  in  contact  with  the  instru- 
ment employed  in  ruling ; when  this  point,  as  in  a flat  ruler, 
cannot  be  very  far  removed  from  the  paper  level,  the  devia- 
tion from  a straight  line  may  not  be  very  perceptible ; but 
when,  as  in  a round  one,  this  point  may  be  an  inch  or  more 
above  the  surface  to  be  ruled  over,  it  wfill  readily,  on  reflection, 
be  seen  that  a very  slight  inclination  of  the  top  of  the  pen, 
either  to  or  from  the  person  using  it,  will  exercise  a con- 
siderable influence  on  the  lower  extremity  also.  One  can, 
in  fact,  draw  two  parallel  lines  some  little  distance  apart 


THE  INKING- IN  OF  LONG  LINES. 


7 


without  shifting  the  ruler  at  all,  simply  by  altering  the  direc- 
tion of  the  holding  of  the  pen. 

13.  The  pressure  of  the  pen  against  the  ruling  surface 
should  be  equable,  or  the  line  Avill  vary  in  thickness;  a 
gradual  or  sudden  compression  of  the  nibs  will  reveal  itself 
in  the  gradual  or  sudden  diminution  of  the  breadth  of  the 
line.  Those  of  maturer  experience  can,  if  they  choose,  pro- 
duce this  effect  when  the  nature  of  the  work  calls  for  it,  but 
with  the  beginner  the  result  is  more  likely  to  be  involuntary 
and  the  effect  undesirable. 

14.  It  is  often  a very  aggravating  circumstance  to  note 
that  the  long  line  is  rapidly  reducing  the  quantity  of  ink 
held  by  the  pen,  and  it  becomes  a matter  of  some  little  im- 
portance as  to  whether  the  supply  will  bold  out  to  the  end 
or  not.  It  is  very  awkward  to  have  to  stop  and  replenish 
the  pen  in  the  middle  of  a line,  as  it  is  decidedly  difficult  to 
go  on  as  though  nothing  untoward  had  happened,  for  the  line 
will  almost  always  betray  the  fact.  The  only  real  remedy, 
when  such  a termination  is  clearly  impending,  is  to  boldly 
stop  and  refill  the  pen  some  time  before  it  gets  exhausted. 
If  the  point  of  exhaustion  has  been  reached,  the  point  where 
the  line  has  been  resumed  will  always  be  perceptible.  One 
soon  learns  by  experience  how  far  one  filling  of  the  pen  will 
carry,  and  when  that  knowledge  has  once  been  gained,  it  is 
only  want  of  reasonable  care  that  leads  to  attackitig  a thing 
with  insufficient  means.  We  need  scarcely  say  that  the  pen 
needs  much  more  rapid  replenishment  when  the  lines  for 
which  it  is  used  are  bold  and  broad  than  it  does  for  finer 
work.  Care  must  be  taken  not  to  overload  the  pen  with  ink, 
or  a big  round  blot  as  soon  as  the  instrument  touches  the 
paper  will  be  the  penalty. 

15.  On  commencing  work  the  pen  should  always,  after  it 
has  been  supplied  with  ink,  be  tried  first  of  all  on  a piece  of 


8 


MATHEMATICAL  INSTRUMENTS. 


waste  paper.  It  is  not  always  easy  to  at  once  produce  the 
right  thickness  of  line,  and  the  preliminary  endeavours  in 
this  direction  may  well  be  spent  on  a spare  fragment  of 
paper  rather  than  on  the  drawing.  This  piece  of  paper, 
however,  should  not  be  too  unlike  the  sheet  on  which  the 
drawing  is  being  made,  for  often  a pen  line  that  will  appear 
on  trial  on  a piece  of  smooth  writing-paper  to  be  just  what  is 
wanted  will,  on  application  to  the  drawing-paper,  produce  a 
very  different  looking  result.  We  dwell  on  this  because  we 
have  so  often  found,  on  advising  our  students  to  try  their 
pens  first  on  a piece  of  waste  paper,  that  their  first  impulse 
has  been  to  dive  into  their  pockets  for  a letter.  Even  the 
failures  of  the  novice  may  be  to  some  extent  utilised,  for 
when  a damaged  drawing  is  torn  up,  its  pieces  form  the 
very  things  we  want  for  trying  the  pens  on  for  the  next. 
Where  the  drawing  is  large,  and  an  ample  margin  can  be 
secured,  it  is  often  the  custom  to  try  the  pens  and  tints  of 
colour,  &c.,  all  round  the  edges,  but  the  beginner,  at  least,  v/ill 
do  well  to  look  upon  his  drawing  as  a thing  to  be  treated  with 
more  respect,  and  to  be  maintained  scrupulously  clean.  A 
blot  of  ink  or  spot  of  wet  colour  on  the  margin  of  the  draw- 
ing may  get  on  the  cuff  of  the  coat,  and  will  thence  very 
readily  be  transferred  to  the  centre  of  the  drawing.  A further 
disadvantage  in  using  the  margin  of  the  drawing  as  a space 
to  try  pens,  &c.,  on,  is  that  very  possibly  some  one  or  more  of 
these  trial  lines  may  encroach  too  far.  Many  a time  have  we 
seen  the  margin  of  a drawing  cut  down  far  too  much  because 
somewhere  a mark  or  two  had  necessitated  the  whole  line  on 
that  side  being  brought  in,  and  that  in  turn  obliged  the  other 
side  to  be  brought  in,  or  the  work  would  have  been  thrown 
out  of  the  centre  of  the  sheet.  This  difficulty  may  neverthe- 
less be  fairly  met  by  at  once,  on  the  very  commencement  of  the 
work,  ruling  a clean  pencil  line  all  found ; all  within  this  is 


HOW  TO  CLEAN  THE  PEN  AFTER  WORK. 


9 


part  of  the  permanent  result,  and  all  behind  it  may  be  used 
for  trial  lines  and  tints.  This  may  more  readily  be  done,  and 
becomes  more  justifiable,  when  a drawing  is  strained,’’  as  it 
is  termed,  or  fastened  down  all  round  its  edges  to  the  board 
by  means  of  glue  and  paste,  for  this  strip  all  round  will,  in 
any  case,  be  discarded  when  the  drawing  is  completed  and 
cut  from  the  board. 

1 6.  All  the  tools  used  should  be  kept  scrupulously  clean; 
all  rulers  that  go  against  paper  should  be  carefully  dusted ; 
all  pens  should  be  carefully  cleaned  when  work  is  over  for 
the  day;  all  brushes  should  have  the  colour  thoroughly 
washed  out  of  them.  A drawing  that  is  not  covered  up  or 
put  away  will,  in  a short  time,  get  a sufficient  amount  of  fine 
dust  on  it  to  affect  the  working  of  the  ruling-pen,  and  no 
work  worth  anything  can  be  done  if  the  ruling-pen  itself 
and  the  inking  compasses  be  not  kept  thoroughly  clean. 
When  through  any  oversight  the  ink  is  left  to  dry  in  a pen, 
it  should  be  removed  by  very  gently  inserting  the  blade 
of  a small  penknife  and  scraping  it  away.  If  the  nibs  are 
close  together,  they  must  be  opened  ; for  the  blade,  if  it  be 
forced  out  between  them  at  the  bottom,  strains  the  regulating 
screw,  and  this  will  prevent,  sooner  or  later,  its  efficient 
action.  ISTo  pen  should  really  be  left  in  this  condition  at  all, 
for  a few  minutes  at  the  end  of  the  work  may  be  excellently 
well  spent  in  cleaning  all  up,  and  putting  everything  out  of 
harm’s  way.  Any  kind  of  soft  rag  or  blotting  paper  will  do 
to  remove  the  bulk  of  the  ink  from  the  nibs,  care  only  being 
taken  that  the  material  employed  will  not  leave  any  hairs  or 
fibres  behind  it : after  this  the  inner  surfaces  of  the  nibs 
may  be  very  well  cleaned  by  doubling  a piece  of  good  stiff 
paper  into  the  shape  of  a V,  and  inserting  it  between  them. 
The  paper  should  not  be  doubled  too  hard ; the  sides  of  the 
V should  retain  a good  deal  of  elasticity,  as  they  are  then 


lO 


MATHEMATICAL  INSTRUMENTS. 


pressed  against  the  inner  sides  of  the  pen  when  the  paper 
is  rubbed  up  and  down.  Many  people  wipe  their  pens  on 
their  cuffs  or  the  inside  of  their  coats,  but  order,  neatness, 
and  cleanliness  are  so  essential  to  success  in  instrumental 
drawing,  that  the  balance  of  success  will  always  tend  towards 
those  who  practise  those  virtues,  and  against  those  who 
ignore  them. 

17.  The  ruling-pen  is,  perhaps,  the  instrument  on  which 
success  or  failure  most  depends,  seeing  that  it  is  in  use  almost 
from  beginning  to  end  of  a drawing : it  is,  therefore,  the  one 
that  will  probably  be  the  first  to  give  out.  'No  time  should 
be  lost  in  replacing  it  if  it  is  hopelessly  done  for,  or  in  getting 
it  reset  if  it  has  simply  got  worn.  Cheap  instrument  sets 
rarely  contain  good  pens;  in  some  they  are  too  weak,  in 
others  the  nibs  are  of  unequal  length.  In  this  latter  case  a 
little  judicious  rubbing  down  on  a stone  will  often  put  mat- 
ters right.  A good  drawing-pen  should  cost  about  two  or 
three  shillings,  the  price  varying  a little  according  to  the 
pattern,  whether  made  of  steel  or  electrum,  and  so  on,  while 
an  old  pen  will  be  reset  by  any  maker  of  instruments  for 
about  threepence.  A little  practice  on  the  part  of  the  owner 
will  soon  enable  him  to  be  independent  of  this  latter  aid : it 
is,  in  any  case,  well  that  he  should  be  able  to  at  once  restore 
a blunted  point,  or  rub  down  one  that  cuts  into  the  paper. 
A long  delay  and  hindrance  is  thus  often  saved  ; in  our 
own  case,  the  nearest  instrument  maker  is,  we  believe,  some 
thirty  miles  off.  A piece  of  common  slate  and  a little  water 
will  often  be  sufficient  to  do  all  that  is  needed;  if  oil  be 
used,  great  care  must  be  taken  that  all  trace  of  it  is  removed 
before  work  is  resumed.  By  using  water,  the  success  attend- 
ing the  setting  can  be  from  time  to  time  tested,  and  any 
defects  that  show  themselves  on  trial  rectified. 

18.  In  choosing  a pen,  one  with  good  stiff  nibs  should 


THE  PRICKER  AND  DOTTING  WHEEL, 


II 


be  selected,  so  that  a moderate  amount  of  pressure  will  not 
close  them  during  use.  A further  desirable  feature  is  seen 
in  some  pens  where  the  lower  part  of  the  handle  is  made 
square ; this  gives  at  once  a better  hold,  and  also  indicates 
the  right  direction  for  the  points. 

19.  In  using  the  drawing-pen,  it  should  be  held  almost 
upright,  and  only  Indian-ink  should  ever  be  employed; 
common  ink  corrodes  the  fine  points  and  surfaces,  and  soon 
damages  the  efficiency  of  the  instrument.  There  is,  of  course, 
no  objection  to  the  use  of  Prussian  blue  or  crimson  lake, 
when,  as  in  some  complex  perspective  or  other  diagrams,  it 
is  desirable,  as  we  have  seen,  to  distinguish  various  sets  of 
lines.  The  only  practical  difficulty  is  in  insuring  a sufficient 
cleanliness  in  the  pen  ordinarily  used  for  black  alone  to  pre- 
vent its  sullying  the  brightness  of  the  colours.  Where  black 
and  red  or  black  and  blue  lines  are  being  used  in  the  same 
drawing,  it  is  a great  saving  of  time  and  temper  to  keep  one 
pen  specially  for  each. 

20.  In  some  pens  the  upper  part  of  the  handle  is  remov- 
able, and,  when  unscrewed,  can  be  used  as  a ''  pricker ; ’’  the 
use  of  this  we  shall  illustrate  further  on  ; while  in  others  the 
unscrewing  reveals  a set  of  little  circular  discs  of  steel  with 
edges  variously  toothed  and  notched.  These  are  called  dot- 
ting-wheels  ; they  can  be  inserted  at  the  bottom  of  the  nib, 
the  pen  is  then  filled  with  ink,  and  the  theory  is  that,  as  they 
revolve,  they  make  dots  or  dashes,  but  practically  the  work 
is  too  coarse  and  rough  to  be  of  any  use,  and  not  unfrequently 
the  first  dot  is  a big  blot,  and  all  the  others  mere  dents  in 
the  paper.  Hand-dotting  is  far  preferable. 

21.  Having  now  dwelt  at  some  little  length  on  what  we 
may  term  the  typical  ruling-pen,  we  proceed  to  give  some 
short  explanation  of  some  of  the  various  modifications  of 
form  that  may  be  met  with. 


12 


MATHEMATICAL  INSTRUMENTS. 


22.  The  bordering-pen,  so  called  from  its  special  adap- 
tability for  making  the  broad  borders  that  are  sometimes 
put  round  large  show  architectural  or  engineering  drawings, 
is  something  like  the  ordinary  pen  in  appearance,  but  larger 
in  its  parts.  It  often  has  a central  tongue  running  all  down 
it,  and  almost,  but  not  quite,  equal  in  length  to  the  nibs. 
By  this  means,  a large  body  of  ink  can  be  retained  for  use 
in  the  pen,  and  a good  broad  line  can  be  made  without 
danger. 

23.  The  road-pen  is  a very  useful  adaptation  of  the  ordi- 
nary form.  It  consists  of  two  pens  placed  side  by  side  and 
joined  together  into  one  handle ; a screw  between  them 
enables  them  to  be  adjusted  to  the  required  distance  apart. 
Such  a pen  is  of  great  service  when  a good  many  parallel 
lines  of  similar  width  have  to  be  drawn ; as,  for  example, 
the  joists  supporting  a floor  or  roof,  the  lines  of  the  roads  in 
a topographical  survey,  or  the  metals  of  a railway.  The 
price  of  a double  pen  of  this  character  would  be  about  six 
shillings. 

24.  The  only  other  modification  of  the  ruling-pen  to  which 
we  need  here  refer  is  that  known  as  the  section-pen.  Where 
a large  surface  has  to  be  covered  over  with  a series  of  equi- 
distant lines,  as  in  the  representation  of  anything  in  section, 
for  example,  the  lines  are  often  so  close  and  so  numerous 
that  it  would  be  endless  trouble  to  measure  them  and  space 
them  all  out.  In  such  case,  it  is  ordinarily  the  custom  to 
draw  them  by  the  eye,  and  a little  practice  will  readily 
enable  any  one  to  do  so,  but  for  those  who  prefer  a mechani- 
cal help,  the  section-pen  will,  no  doubt,  be  an  assistance, 
though  we  do  not  remember  ever  to  have  seen  one  actually 
in  use,  and  have  certainly  never  ourselves  possessed  one. 
Under  these  circumstances,  while  we  refer  to  it  as  an  instru- 
ment that  our  readers  may  possibly  come  across,  we  can 


PEN  FOR  RULING  SECTION-LINES. 


13 


scarcely  be  expected  to  very  warmly  advise  our  readers  to 
procure  one.  Beyond  the  two  nibs  of  an  ordinary  pen  is  a 
third  that  can  be  adjusted  by  means  of  a screw  to  the  required 
distance  that  the  lines  to  be  drawn  will  be  apart.  This  third 
nib  is  placed  on  the  first  line  drawn  and  run  along  it,  while 
the  pen  portion  is  drawing  line  number  two  : the  nib  is  then 
in  like  manner  placed  in  number  two  while  the  third  line  is 
drawn,  and,  as  the  work  proceeds,  the  lines  must  necessarily 
follow  each  other  at  equal  distance,  and  produce  an  even 
tint.  As  an  instrument  of  this  kind  costs  five  or  six  shil- 
lings, it  will  probably  be  regarded  by  many  of  our  readers 
rather  as  a luxury  than  a necessity. 


( 14  ) 


CHAPTEE  II. 

The  T square — Directions  for  its  use— Must  never  be  cut — Greek  fret- 
pattern — Squaring-off  a drawing — Materials  used  in  making  T 
squares — Useful  size  to  get — Cost — Various  forms  of  T square — 
Stanley  square  — Movable-headed  square  — The  straight-edge — 
Useful  size  to  get — Importance  of  seasoned  wood — Materials  em- 
ployed for  straight-edges — Straight-edge  used  for  dividing  paper — 
How  to  test  accuracy  of  straight-edge — Needles  for  centres  or  vanish- 
ing points — The  marking  of  points. 

25.  Still  confining  ourselves  to  the  means  whereby  we 
may  draw  straight  lines,  we  now  proceed  to  consider  the 
nature  and  construction  of  the  T square,  an  indispensable 
item  in  the  equipment  of  the  architect  and  draughtsman. 
The  T square  is  so  called  from  its  resemblance  in  shape  to  the 
letter  T ; the  long  part  is  termed  the  blade,  the  thicker  cross 
piece  the  stock.  Its  possession  implies  the  acquisition  of  a 
drawing-board,  as  the  square  is  of  no  use  without  it.  The 
stock  or  head  of  the  T square  is  pressed  against  one  edge  of 
the  board,  and  all  lines  then  drawn  will  be  parallel  to  each 
other.  The  same  edge  should  be  used  throughout,  and  the 
square  should  always,  when  its  blade  is  lying  on  the  paper, 
have  its  stock  on  the  left-hand  side  of  the  draughtsman. 
Some  beginners  are  so  used  to  working  almost  everything 
with  the  right  hand  that  they  endeavour  to  manipulate  the 
square  in  this  way.  In  some  forms  of  square,  as  in  Fig.  2, 
where  the  blade  increases  in  width  towards  the  stock,  this 
would  be  almost  impossible,  and  in  any  case  it  is  very  un- 


METHOD  OF  USING  T SQUARE,  15 


desirable,  whereas  by  working  the  square  into  any  required 
position  by  means  of  the  left  hand,  the  right  is  free  to  retain 
hold  of  the  pencil  or  pen  and  to  at  once  use  them. 

26.  When  the  board  is  tilted  and  the  paper  is  somewhat 
smooth,  the  T square  will  sometimes  gradually  slip  down, 
and  especially  when  the  drawing  is  left  for  awhile,  and  the 
beginnings  of  mischief  are  not  therefore  perceived.  It  will 
be  sufficiently  obvious  that  a fall  from  the  table  to  the  floor 
may  do  the  square  considerable  damage  either  in  splitting 
the  blade  or  loosening  its  attachment  with  the  stock.  To  avoid 
this  the  square  should  either  be  lifted  off  the  drawing  and 
placed  on  the  flat  table  by  the  side  of  the  board,  or  a draw- 
ing pin  may  be  put  somewhere  between  the  square  and  the 
bottom  of  the  board  to  arrest  its  progress  if  it  should  begin 
to  slip.  X squares  when  not  in  use  should  be  hung  up  by 
means  of  the  hole  that  for  that  purpose  is  made  near  the  top 
of  the  blade.  They  will  then  collect  less  dust  and  be  freer 
from  accidental  risks.  A T square  should  always  in  any  case 
be  wiped  before  using,  as  the  dust  collecting  surface  is  a 
large  one,  and  every  pains  should  be  taken  not  to  destroy  the 
cleanliness  of  the  drawing. 

27.  If  the  drawing-board  is  well  made,  if  its  sides  are  true 
right  angles  to  each  other,  a great  saving  of  time  results,  as  by 
placing  the  stock  on  the  left  edge  all  the  horizontal  lines  can 
be  drawn,  and  then  by  placing  it  on  the  bottom  edge  of  the 
board  all  the  necessary  lines  at  right  angles  to  the  first  series 
can  be  quickly  and  accurately  produced.  Such  a method  of 
working  would  be  very  advantageous,  for  example,  in  an  archi- 
tectural elevation,  as  all  the  lines  of  the  cornices,  tops  and 
bottoms  of  windows  and  doors,  lines  of  mouldinirs,  strinof- 
courses,  and  steps,  could  be  drawn  first,  and  then  all  the 
lines  perpendicular  to  these,  the  side  walls  themselves,  the 
uprights  of  all  doors  and  windows,  and  so  on.  We  should  use 


i6 


MATHEMATICAL  INSTRUMENTS. 


it,  again,  in  a plan  where  the  various  walls  were  at  right  angles 
to  each  other.  In  drawing  the  second  series  of  lines  the 
stock  should  always  be  placed  against  the  lower  edge  of  the 
board  and  not  against  the  top.  As  the  square,  to  be  thor- 
oughly useful,  should  always  be  a little  longer  if  anything 
than  the  long  side  of  the  board,  it  will  evidently  be  con- 
siderably longer  than  the  short  side.  By  putting  the  stock 
against  the  lower  edge  the  part  of  the  blade  that  projects 
beyond  the  line  of  the  board  is  quite  out  of  the  way,  but 
beginners  have  often  a perverse  way,  that  we  could  never 
see  a motive  for,  of  placing  the  stock  on  the  top  edge  of  the 
board,  reaching  all  over  their  work  to  regulate  the  move- 
ments of  the  square,  and  having  the  long  overlapping  blade 
coming  out  towards  them  in  a most  embarrassing  way. 
When  the  square  is  a long  one,  they  can  hardly  get  near 
their  work  at  all,  and  any  moment  some  movement  of  their 
body  may  give  the  end  of  the  square  a jerk  that  will  be  very 
detrimental  to  the  appearance  of  any  line  that  may  be  being 
drawn  at  the  time. 

28.  When  we  come  to  describe  the  properties  appertaining 
to  a good  drawing-board,  we  shall  give  a simple  geometrical 
method  by  which  one  can  readily  tell  if  the  angles  are  true 
right  angles  or  not.  The  working  of  the  T square  for  each 
set  of  lines  is  a great  convenience  where  the  lines  are  some- 
what long,  but  in  a good  deal  of  drawing  work  the  ”]"  square 
practically  is  used  only  for  all  the  horizontal  lines,  the 
others,  at  right  angles  to  these,  being  ^ drawn  by  means  of 
another  instrument,  the  set-square,  an  instrument  that  w^e 
shall  describe  in  due  course. 

29.  It  is  always  advisable,  if  possible,  to  use  the  same  T 
square  all  through  the  drawing,  and  never  on  any  considera- 
tion allow  it  to  be  used  either  by  yourself  or  others  to  cut 
off  drawings  from  the  board.  The  T square  may  very 


QF  THE 


' iO 


REPRODUCING  WORKING  DRAWINGS, 


17 


conveniently  be  employed  to  sqnare-off  the  drawing,  and 
these  pencil-lines  will  he  the  guide  for  the  subsequent 
cutting-off ; but  it  can  never  be  too  strongly  impressed  on 
the  novice  that  one  slip  of  the  knife,  making  one  small 
notch  in  the  edge  of  the  blade,  utterly  destroys  the  square 
for  ruling  any  more  straight  lines.  A metal  or  metal-edged 
straight-edge  should  always  be  used  for  all  cutting  work. 

30.  In  inking-in  a large  drawing,  or  one  with  many  lines 
in  it,  it  will  practically  be  found  convenient  to  ink-in  all 
the  lines  in  one  direction  first,  and  then  to  turn  the  atten- 
tion to  the  others  at  right  angles  to  them.  A great  saving 
of  time  is  effected,  as  all  the  time  spent  in  shifting  the  square 
from  one  edge  of  the  board  to  the  other  is  economised. 

3 1 . As  an  exercise  in  the  use  of  the  T square  we  have  given 
in  fig.  I an  illustration  of  a Greek  fret  pattern.  All  the  lines 
can  be  formed  by  the  square  alone.  In  reproducing  it,  divide 
the  whole  space  up  first  into  squares,  as  shown. 

32.  Where  a working  drawing  has  got  so  destroyed  that 
it  is  desirable  to  make  a new  copy  of  it,  this  can  be  very 
expeditiously  done  by  pinning  the  old  drawing  down  on  the 
right-hand  side  of  the  board  and  squaring  off  all  the  lines 
that  run  in  one  direction,  and  then  fastening  it  down  at  the 
top  of  the  board  and  squaring  down  all  the  lines  at  right 
angles  to  the  first.  This  is  a method  often  adopted  in  prac- 
tice where  no  very  great  accuracy  is  called  for,  but  it  is  not  to 
be  commended  to  the  beginner,  as  he  can  only  blindly  copy 
what  he  sees,  and  he  loses  the  inestimable  advantage  of  being 
obliged  to  take  accurate  measurements  step  by  step  through- 
out his  work.  As  it  is,  under  certain  limitations,  one  legi- 
timate use  of  the  T square,  we  here  refer  to  it,  though 
ordinarily  it  is  the  rough  and  ready  way  of  the  workshop, 
where  time  is  money,  and  in  really  careful  work  would  be 
inadmissible. 


B 


l8  MATHEMATICAL  INSTRUMENTS. 


33-  T squares  are  ordinarily  made  either  of  pearwood  or 
mahogany.  Of  these,  the  pearwood  are  preferable,  as  they 
are  not  so  liable  to  warp,  and  the  density  of  the  grain  of  the 
wood  appears  more  dirt-resisting.  We  have  always  ourselves 
found  that  it  was  more  difficult  to  keep  a drawing  clean 
with  a mahogany  square  than  one  of  pearwood.  Mahogany 
gives  an  unpleasant  edge,  such  squares,  therefore,  have 
ordinarily  a line  of  ebony  to  rule  against.  Sometimes  the 
entire  square  is  made  of  ebony,  but  these  cannot  be  recom- 
mended, as  they  collect  dirt,  and,  owing  to  the  deep  colour  of 
the  wood,  do  not  show  it  until  mischief  possibly  is  done. 
In  some  squares,  again,  a line  of  brass  is  let  in  along  the 
ruling  edge  of  the  blade.  Such  squares  can  of  course  be  used 
for  cutting  out  with,  but  they  do  not  ordinarily  last  very 
long.  The  expansion  and  contraction  of  the  metal  soon 
render  its  hold  insecure,  and,  like  many  other  things  that 
are  good  in  theory,  it  fails  to  stand  the  searching  test  of 
practical  use. 

34.  The  square  employed  should  bear  a just  proportion  to 
the  work  .that  is  expected  of  it,  and  should  be  about  equal 
in  length  to  the  length  of  the  drawing-board  with  which  it 
will  be  used.  Where  drawings  of  various  sizes  are  made,  the 
squares  should  err  rather,  if  at  all,  in  being  over  large  for 
some  of  them,  for  the  small  drawings  can  be  worked  by  the 
aid  of  the  square  that  is  used  for  the  larger,  but  the  reverse 
is  by  no  means  the  case ; the  large  drawings  cannot  be  worked 
with  a small  square.  The  happiest  arrangement  in  such  a 
case,  we  need  scarcely  indicate,  is  to  have,  if  possible,  two  or 
three  squares  of  various  lengths.  The  custom  of  the  school  or 
office  in  which  the  draughtsman  finds  himself  will  go  far  to- 
wards settling  the  point,  as  there  is  almost  always  in  such  cases 
some  recognised  standard  of  uniformity  in  the  drawings 
produced,  that  is  as  nearly  as  possible  preserved  all  through. 


VARIOUS  FORMS  OF  T SQUARE. 


19 


35.  In  speaking  of  the  size  of  a square,  the  length  of  the 
blade  exclusive  of  stock  is  always  understood,  so  that  a 36- 
inch  T square  would  not  go  into  a 36-inch  box.  For  dia- 
grams and  large  drawings  T squares  of  60-inches  blade  may 
be  employed,  while  the  draughtsman  on  wood,  or  the  wood 
engraver  into  whose  hands  his  work  passes,  use  squares 
that  are  often  less  than  a foot  long.  Owing  to  this  great 
difference  in  size  and  in  the  material  employed,  it  becomes 
impossible  to  say  in  an  offhand  way  what  the  cost  of  this 
instrument  will  be  to  the  beginner ; but  to  quote  one  price 
alone,  if  the  possession  of  a square  of  pearwood  having  a 
blade  of  some  30  inches  long  be  deemed  about  the  right 
thing,  as  it  ordinarily  would  be,  such  a square  should  cost 
something  between  two  and  three  shillings.  Anything 
between  a shilling  and  a sovereign  may  be  expended. 

36.  Squares  vary  in  the  details  of  their  construction,  and 
to  some  of  these  modifications  we  now  proceed  to  allude. 

37.  The  old  form  of  T square  had  the  blade  let  into  the 
stock,  the  blade  itself  was  of  one  width  throughout,  and  the 
stock  had  a rabbet  that  fitted  to  the  edge  of  the  board,  and 
this  is  still  the  accepted  form  with  many  English  makers, 
and  remains  the  accepted  type  on  the  Continent.  It  is  re- 
presented in  figs.  16  and  17.  What  is  termed  the  Stanley 
square  is  a great  improvement  in  many  ways  on  this.  It 
derives  its  name  from  the  maker,  one  of  our  most  successful 
practical  purveyors  of  the  various  instruments  required  for 
the  draughtsman.  We  have  no  interest  in  any  way  in  com- 
mending the  wares  of  any  one  man  over  those  of  others,  but, 
as  our  book  is  intended  to  be  of  really  practical  value  to 
those  seeking  aid,  it  is  false  delicacy  to  refrain  from  this 
mention  of  names.  The  Stanley  square,  fig.  2,  is  merely  a 
blade  of  pearwood  or  mahogany  screwed  on  a stock ; it  stands, 
therefore,  above  the  stock.  This  form  of  construction  pre- 


20 


MATHEM)ATICAL  INSTRUMENTS, 


sents  several  advantages  that,  in  practical  work,  are  readily 
perceived.  One  of  these  is,  that  owing  to  the  blade  being 
above  the  stock,  the  set-square  passes  freely  on  to  it,  and 
allows  of  lines  being  ruled  right  to  the  edge  of  the  board. 
Another  advantage  is  that  in  case  of  any  accident,  such  as 
notching  the  edge,  the  blade  can  readily  be  removed,  and  the 
damage  set  right,  and  then  all  screwed  on  again.  In  the 
old  form  of  square  it  would  be  impossible  to  do  this.  The 
blade,  too,  tapers  considerably,  it  is  much  wider  at  the  stock 
end  than  the  other.  This  form  gives  great  strength,  and  pre- 
vents the  deflection  of  the  end  of  the  blade. 

38.  Another  kind  of  T square  has  a movable  head,  so  that 
the  blade  can  be  set  to  any  required  angle  with  the  stock, 
and  then  screwed  flrmly.  Such  a construction  is  valuable 
where  a great  many  parallel  oblique  lines  have  to  be  drawn, 
but  in  practice  there  is  not  much  demand  for  a square  of  this 
kind,  as  a judicious  use  of  the  ordinary  T square  and  set- 
square  combined  will  ordinarily  meet  every  want.  It  must 
be  borne  in  mind,  too,  that  a movable  head  and  screw,  and 
so  forth,  all  mean  more  work,  and,  consequently,  more 
expense. 

39.  The  straight-edge,  represented  in  flgs.  ii,  12,  13,  is 
another  exceedingly  useful  item  in  the  equipment  of  the 
draughtsman.  It  is  merely  a long  thin  strip  of  wood,  its 
sides  parallel,  and  its  breadth  and  thickness  depending  on 
the  length.  The  length  is  a very  variable  quantity:  for 
small  drawings,  a straight-edge  of  some  1 8 or  20  inches  long 
is  very  useful,  but  in  large  perspective  drawings,  where  the 
vanishing  points  may  be  10  feet  or  more  away  from  the 
board,  the  straight-edge  employed  must  be  at  least  as  long. 
A 24  - inch  straight  - edge  should  cost  about  a shilling, 
while  a lo-footer  would  be  about  half  a guinea.  We  have 
ourselves  repeatedly  used  straight-edges  that  could  not  even 


THE  STRAIGHT-EDGE, 


21 


be  got  into  many  rooms  at  all.  Both  edges  are  planed  to  a 
true  straight  line,  and  often  in  the  larger  and  thicker  instru- 
ments one  of  these  edges  is  bevelled,  so  as  to  render  it  more 
convenient  for  the  ruling-pen. 

40.  Wood  and  steel  are  the  materials  employed.  Where 
wood  is  used,  care  should  be  taken  to  see  that  it  is  thoroughly 
seasoned,  for  if  it  is  not,  such  long  thin  strips  have  a great 
tendency  to  warp  and  twist  from  the  true  line.  This,  how- 
ever, is  really  a matter  for  the  maker  to  see  to,  the  purchaser 
can  only  take  what  is  given  to  him,  and  wait  till  time  proves 
either  the  value  or  the  worthlessness  of  his  purchase.  It  is 
well,  therefore,  to  go  for  all  things  of  this  kind  to  a man 
who,  having  made  a name,  has  a name  to  lose.  Such  pur- 
chases do  not  often  need  to  be  made,  and  economy  is  better 
studied  by  paying  a little  more  for  a really  good  thing  than 
by  buying  so-called  cheap  things — things  that  will  probably 
be  a constant  annoyance,  hardly  bad  enough  to  throw  away, 
hardly  good  enough  to  keep.  A straight-edge  now  hanging 
in  our  room  is,  we  see,  dated  1 867 ; it  is  as  sound  and  true 
as  it  ever  was,  has  been  freely  used  ever  since  it  was  bought, 
and  will  serve  us  for  as  many  years  to  come  as  we  are  likely 
to  call  upon  it  to  do. 

41.  In  some  straight-edges  two  or  three  pieces  of  wood 
are  glued  together  to  counteract  any  disposition  towards 
warping,  and  in  others  the  centre  is  of  one  kind  of  wood  and 
the  two  strips  that  form  the  edges  are  of  another,  the  idea 
in  this  case  also  being  to  prevent  twisting.  A straight-edge 
should  be  of  sufficient  breadth  and  thickness  in  proportion  to 
its  length  to  give  rigidity.  Steel  gives  an  admirably  true 
line;  but  straight-edges  of  this  material  are  about  seven  times 
as  costly  for  any  given  length  as  wooden  ones.  A steel 
straight-edge  has  the  great  recommendation  that  it  can  also 
be  freely  used  when  cutting  out  drawings  has  to  be  done,  for 


22 


MATHEMATICAL  INSTRUMENTS. 


we  need  scarcely  say  that  a notch  on  the  edge  of  a wooden 
straight-edge  spoils  it  as  effectually  as  it  does  a T square, 
and  all  the  warnings  that  we  have  already  given  might  well 
be  repeated  here.  Steel  has,  however,  one  or  two  great  dis- 
advantages. It  is  much  heavier  in  proportion  than  wood, 
the  moisture  of  the  hand,  too,  is  likely  to  rust  it,  and  this 
rust  gets  on  the  drawing,  and  it  is,  in  addition,  very  cold  to 
use  for  a great  part  of  the  year.  When  the  temperature  is 
low  the  paper  and  instruments  are  all  very  cold  to  the  touch, 
and  cold  hands  are  a terrible  hindrance  to  good  work. 

42.  When  it  is  required  to  divide  up  sheets  of  paper,  the 
work  can  be  well  and  expeditiously  done  by  placing  the  steel 
straight-edge  at  the  line  of  required  division,  holding  it  firmly, 
and  then  taking  the  paper  by  one  corner,  and,  while  slightly 
raising  it,  tearing  it  sharply  down.  The  edge  is  almost  as 
sharp  as  if  cut  by  a knife.  A quire  or  ream  of  paper  may  in 
this  way  be  divided  up  into  half  or  quarter  sheets  in  far  less 
time  than  by  any  process  of  folding  or  cutting.  We  place 
the  paper  squarely  on  a drawing-board,  find  out  by  measur- 
ing one  sheet  where  the  true  division  should  come,  make  two 
marks  on  the  board,  one  above  and  one  below  the  sheets  of 
paper  at  that  point,  place  the  straight-edge  truly  by  these, 
and  then  tear  away  sheet  after  sheet  without  fear  or  failure. 

43.  Some  straight-edges  have  inches  or  feet  marked  on 
them,  but  the  advantage  of  this  is,  in  most  cases,  not  very 
great.  A straight-edge  is  more  ordinarily  a means  of  ruling 
than  of  measuring  a line.  In  the  same  way  some  T squares 
have  inches  marked  on  them,  but  all  such  dimensions  can 
more  readily  be  found  on  the  scales  that  accompany  every 
set  of  instruments.  There  is  always  a strong  risk  that  when 
any  one  thing  is  set  to  two  or  three  different  uses,  it  in  some 
degree  fails  in  doing  any  of  them  effectively. 

44.  The  straight-edges  of  any  respectable  maker  should  be 


TESTING  OF  THE  STRAIGHT-EDGE, 


23 


above  reproach,  but  if  the  owner  of  one  has  his  suspicions,  a 
very  effective  way  of  testing  the  matter  is  to  make  two 
marks  about  the  length  of  the  implement  apart  on  a sheet  of 
paper ; the  straight-edge  should  then  be  placed  to  these,  and 
a line  very  carefully  drawn  by  its  aid  from  one  to  the  other. 
The  straight-edge  should  now  be  turned  over  and  again  placed 
to  these  marks,  and  a second  line  drawn  from  one  to  the 
other.  If  one  line  effectually  covers  the  other  from  end  to 
end,  it  is  a sufficient  indication  that  the  straight-edge  is  true. 
If  it  deviates  at  any  given  point  the  tenth  of  an  inch  from 
the  true  straight  line,  the  deviation  would  show  very  con- 
spicuously when  the  second  line  was  drawn,  as  there  would 
then  at  that  point  be  an  opening  of  a fifth  of  an  inch  between 
the  two  lines  that  should  be  coincident,  and  the  eye  would 
readily  detect  the  error. 

45.  Where  one’s  instruments  are  exposed  to  the  risk  of 
meddling,  or  the  nuisance  of  careless  borrowing,  it  is  w^ell 
just  before  using  the  straight-edge  to  run  a finger  along  the 
edge ; any  little  cut  or  dent  will  be  by  this  means  readily 
appreciated,  or  we  should,  perhaps,  rather  say  ascertained,  for 
appreciation  in  the  ordinary  sense  of  the  word  is  scarcely 
the  feeling  with  which  we  greet  the  damage  done  to  our 
property. 

46.  When  we  have  occasion  to  rule  many  lines  that  all 
go  to  or  towards  one  point,  it  is  a great  saving  of  time  and 
patience  to  put  a needle  into  the  board  at  this  point.  The  edge 
of  the  straight-edge  or  set-square  is  then  kept  in  contact  with 
this,  and  one  end  of  our  required  line  is  thus  always  found, 
and  we  have  only  to  make  sure  of  one  point  instead  of  two 
each  time.  In  drawing  a wheel  of  fifty  teeth,  a portion  of 
the  line  of  each  side  of  each  tooth  would,  if  continued,  run  to 
the  centre.  If,  then,  having  marked  off  on  the  pitch  line  of 
the  wheel  the  starting  points  of  these  lines,  we  place  a needle 


24 


MATHEMATICAL  INSTRUMENTS, 


in  tlie  centre,  the  hundred  lines  will  be  far  more  expedi- 
tiously done  than  if  we  had  each  time  to  place  the  straight- 
edge to  both  centre  and  pitch  line.  We  find  the  same  thing 
available  in  a perspective  drawing;  a needle  placed  at  a 
vanishing  point  is  in  the  same  way  a great  assistance.  In 
very  largo  drawings,  where  the  straight-edges  may  be  mea- 
sured by  feet,  a good  stout  bradawl  firmly  fixed  in  the  table 
takes  the  place  of  the  needle. 

47.  Where  a point,  from  which  a line  has  to  be  drawn  is 
already  somewhere  in  another  line,  and  therefore  hardly 
noticeable,  or  when  from  any  other  reason  it  fails  to  catch 
the  eye  readily,  it  is  a good  plan  to  take  the  pencil  and 
roughly  draw  a little  circle  round  it,  the  size  possibly  of  a 
threepenny-piece.  The  eye  then  readily  detects  its  position. 
Where  several  points  require  this,  it  will  be  easily  seen  that 
various  methods  of  making  them  must  be  adopted ; a small 
square  may  be  placed  round  one,  while  another  may  have 
two  lines  at  right  angles,  a little  cross,  crossing  each  other  at 
the  point  as  their  centre.  When  some  few  lines  have  been 
drawn  actually  to  the  point,  its  position  is  sufficiently  defined, 
but  in  many  cases,  as  in  the  teeth  of  the  wheel,  though  the 
lines  would  go  to  the  point  if  they  were  continued,  they 
never  are  continued,  and  so  the  position  of  the  point  therefore 
remains  possibly  somewhat  hard  to  find. 

48.  Lines  should  always  be  drawn  from  left  to  right,  and 
where  a certain  point  marks  one  extremity  of  a line,  always, 
if  possible,  let  that  be  the  starting-place.  The  work  that 
starts  from  a dot  or  point  is  more  likely  to  be  accurate 
than  the  line  that  goes  to  a point.  If,  for  example,  we  have 
a square,  and  we  want  to  divide  it  into  six  equal  strips  by 
horizontal  lines,  we  should  measure  off  the  divisions  and 
mark  off  the  true  startingrpoints  on  the  left-hand  line,  because, 
in  any  case,  we  should  draw  our  lines  from  left  to  right,  and 


EXPERIENCIA  DOCET, 


25 


our  work  would  be  more  likely  to  be  accurate  if  we  drew 
from  a point  well  in  sight  than  towards  one  some  little  dis- 
tance off,  and  probably  hidden  by  our  right  hand.  Such 
directions  may  not  appear  of  any  great  moment,  but  they 
are  the  result  of  some  years  of  experience,  and  it  is  generally 
in  the  knowledge  of  some  hundreds  of  little  things  rather 
than  in  a few  great  ones  that  the  difference  between  the 
beginner  and  the  older  hand  is  found. 


( 26  ) 


CHAPTEE  III. 

The  set-square — Its  form  and  nature — Useful  size  to  get — Warping, 
and  how  to  remedy  it — Materials  employed — Framed  set-squares — 
Cost — How  to  use  the  set-square — Section  lines — The  set-square  of 
45° — The  set-square  of  6o° — Drawing  nuts — Geometrical  means  of 
testing  the  correctness  of  the  set-square— Parallels  and  perpendicu- 
lars— Batter  lines — Earth  slopes — Hoof  pitches — Parallel  ruler — 
Boiling  parallels — Cost. 

49.  Useful,  and  in  fact  indispensable  to  many  kinds  of  draw- 
ing as  the  y square  and  straight-edge  are,  we  proceed  now  to 
give  some  brief  account  of  the  set-square,  an  instrument  of  per- 
haps even  greater  value,  as  by  the  aid  of  two  set-squares  alone 
we  can,  without  either  of  the  implements  we  have  referred 
to,  or  a drawing-board,  draw  any  number  of  lines  parallel 
to  each  other,  at  right  angles  to  each  other,  or  making  various 
angles.  They  are  therefore  very  useful  where  the  work  is 
small,  as  in  geometrical  problems,  or  where,  owing  to  the 
drawings  being  in  a notebook,  it  is  impossible  to  avail  our- 
selves of  the  aid  of  the  T square.  How  these  various  results 
are  effected  it  will  be  our  duty  to  explain ; but  we  first  give 
some  few  introductory  remarks  as  to  the  nature  of  the  set- 
square,  its  forms,  its  material,  its  cost,  and  then  we  proceed 
to  illustrate,  as  far  as  may  be,  the  various  methods  of  its 
use. 

50.  The  set-square  is  a triangular  piece  of  wood,  vulcanite, 
or  metal.  One  of  its  angles  is  always  90°,  or  a right 


CORRECTION  OF  WARPING. 


27 


angle,  while  the  other  angles  are  ordinarily  either  both  45®, 
or  one  of  them  is  60''  and  the  other  30®.  The  student  will 
he  careful  to  provide  himself  with  one  of  each  of  these ; they 
are  ordinarily  called  ''45  set-squares  and  “60”  set-squares. 
They  vary  a good  deal  in  size,  but  a ''  45  ” of  six  inches  in 
length  is  a very  useful  size,  and  would  cost  in  pearwood 
about  sixpence,  and  in  vulcanite  as  much  again ; while  the 
‘'60''  should  be  what  is  called  a 10-inch,  and  will  cost 
about  eightpence,  or  as  much  again,  according  to  the  ma- 
terial. 

5 1.  The  ordinary  pear-tree  set-squares  are  very  cleanly  and 
pleasant  in  use,  but  there  is  some  little  risk  of  their  curling 
and  warping  by  heat  or  damp,  and  where  this  is  past  remedy, 
the  squares  are  spoilt,  as  the  truth  of  the  angles  is  destroyed, 
and  the  edges  no  longer  remain  straight  lines.  With  reason- 
able care,  however,  such  set-squares  will  prove  all  that  the 
draughtsman  can  wish.  When  one  of  these  curls  with  the 
heat,  either  by  being  too  near  the  fire,  or  from  being  on  a 
table  with  the  sunshine  on  it,  the  immediate  prospect  is  not 
encouraging ; but  if  the  set-square  be  at  once  turned  over  on 
the  other  face,  it  will  then  begin  to  go  back,  and  would  pre- 
sently be  quite  as  bad  as  before.  All  that  is  necessary  is  to 
keep  an  eye  on  it  till  the  back  curling  has  reached  a point 
when  the  instrument  is  straight  again,  and  then  to  at  once 
remove  it  from  the  heat. 

52.  Some  set-squares  are  built  up  of  three  narrow  strips 
of  hard  wood,  generally  mahogany,  and  edged  with  ebony. 
Such  set-squares  are  lighter  than  those  made  in  one  solid 
piece,  and  they  have  not  the  same  tendency  to  warp,  but  the 
ordinary  kinds  are  very  comfortable  to  use,  and  the  little 
additional  lightness  of  the  framed  set-squares  is  no  great 
advantage  practically.  Another  advantage  claimed  for  the 
framed  set-squares  is,  that  they  obscure  so  much  less  of  the 


28 


MATHEMATICAL  INSTRUMENTS. 


drawing  beneath  them  than  the  solid  ones  do,  but  this  again 
is  practically  no  great  gain,  as  the  breadth  of  the  framing 
is  necessarily,  for  the  sake  of  strength,  sufficiently  great  to 
obscure  all  in  the  immediate  neighbourhood  of  the  line  being 
drawn,  and  to  see  quite  clearly  a triangular  space  of  the  draw- 
ing some  two  inches  or  so  from  this  is  no  very  tangible  ad- 
vantage. These  framed  set-squares  are  necessarily  very  much 
more  trouble  to  make,  as  the  angles  at  which  the  pieces 
are  cut  must  be  very  true,  and  these  pieces  when  cut  must  be 
fastened  together  by  metal  tongues  and  rivets.  Where  large 
set-squares  are  required,  and  where  they  will  be  exposed  to 
great  variations  of  temperature,  as  in  India,  the  framed 
pattern  is  undoubtedly  the  best.  They  are  fully  six  times 
the  price  of  the  plain  pearwood  patterns. 

53.  Set-squares  are  now  sometimes  made  of  vulcanite,  a 
preparation  of  india-rubber.  These  are  very  smooth  to  the 
touch,  are  far  less  likely  to  break  by  a fall  or  by  being 
trodden  on,  can  be  washed  if  they  get  soiled,  and  as  they  do 
not  warp,  there  is  no  risk  of  their  angles  or  sides  getting  out 
of  truth.  The  only  objections  that  we  ourselves  feel  to  them, 
and  our  objections  do  not  outweigh  their  good  qualities,  are 
that  they  are  very  cold  to  work  with,  and  that  an  accidental 
spot  of  ink  upon  them  does  not  show,  as  they  are  as  black  as 
ink  themselves,  and  such  a spot  may  very  readily  be  trans- 
ferred by  the  set-square  to  the  drawing.  They  are  about  as 
expensive  again  as  the  solid  pearwood  instruments. 

54.  Every  straight-edge,  T square,  and  set-square  (except 
the  framed  kind),  has  a round  hole  in  it  to  hang  it  up  by, 
and  all  these  instruments  are  the  better  for  this  round  hole 
being  applied  to  its  proper  use.  Instruments  that  are  about 
on  desks  and  tables  catch  much  dust  and  are  exposed  to  divers 
risks  that  those  that  are  hung  up  escape.  Boys  find  an  ever 
new  delight  in  putting  their  fingers  or  pencils  through  this 


THE  NATURE  OF  SECTION  LINES, 


29 


hole  in  their  set-sqnare  and  setting  the  instrument  spinning, 
but  this  we  need  scarcely  say  was  not  the  original  intention. 

55.  By  means  of  the  ‘‘45  set-square  we  can  readily  draw 
lines  perpendicular  to  each  other,  or  lines  that  form  half  a 
right  angle,  i.e.,  45°.  In  fig.  2 we  have  a sufficiently  clear 
illustration  of  how  this  may  be  done.  When  the  set-square 
has  one  of  its  edges  resting  on  the  T square,  the  other  two 
edges  will  either  be  at  right  angles  to  this  and  at  45°  respec- 
tively, as  shown  in  the  set-square  that  is  nearest  the  stock  of 
the  T square,  or  it  may  be  so  placed  that  both  the  edges 
away  from  the  T square  shall  be  at  45°  with  it.  One  of  these 
lines  will  then  slant  upwards  to  the  right  and  the  other  up- 
wards to  the  left ; this  may  easily  be  seen  by  referring  to  the 
second  set-square  shown  in  our  figure.  The  student  will 
have  no  difficulty  in  copying  the  geometrical  patterns  we  have 
indicated  in  our  illustration.  It  will  also  readily  be  perceived 
that  a square  may  be  drawn  without  any  aid  of  compasses  or 
any  geometrical  method.  This  is  shown  in  the  small  figure 
above  the  patterns.  A line  is  first  drawn  of  the  required 
length  as  a base,  and  then  at  each  extremity  of  it  lines  are 
drawn  at  an  angle  of  45°  with  it ; perpendiculars  from  each 
end  of  the  base  line  will,  where  they  intersect  these  oblique 
lines,  give  the  remaining  angles  of  the  figure. 

56.  The  set-square  of  45°  is  also  very  largely  used  for 
another  purpose,  that  of  marking  in  an  architectural  or 
engineering  drawing  what  parts  of  the  objects  there  shown 
are  intended  to  be  what  is  technically  termed  ''  in  section.” 
This  term  is  applied  to  any  portion  that  is  supposed  to  be 
cut  through,  and  these  cuttings  or  sections  are  largely  used 
in  things  of  complex  construction,  as  by  their  means  one  is 
enabled  to  learn  a good  deal  of  the  interior  parts  and  those 
portions  that  are  obscured  by  reason  of  others  being  in  front 
of  them.  In  an  architectural  drawing,  for  example,  a section 


30 


MATHEMATICAL  INSTRUMENTS, 


shows  a good  deal  that  could  never  be  gathered  from  the 
other  drawings.  We  have  in  fig.  3 taken  a very  simple 
illustration.  We  have  cut  through  a box,  and  it  will  be  seen 
on  a short  reflection  that  our  only  way  of  knowing  how  thick 
the  woodwork  was  would  be  by  this  section  through  it. 
Neither  side  view,  end  view,  nor  plan  would  give  this  infor- 
mation. In  fig.  4 we  have  taken  a longitudinal  section 
through  a piece  of  piping.  When  a section  is  taken  in  the 
direction  of  the  breadth  it  is  termed  a transverse  or  cross 
section.  When  it  is  taken  in  the  direction  of  the  length  of 
the  object  the  section  is  called  longitudinal. 

57.  When  a section  through  an  object  reveals  the  fact  that 
it  is  composed  of  different  substances,  this  difference  is  shown 
by  taking  the  section  lines  in  different  directions,  though 
always  at  45°.  This  arrangement  is  indicated  in  fig.  5.  Our 
readers  can  readily  realise  what  we  mean  by  the  simple 
illustration  of  what  we  should  see  on  splitting  a blacklead 
pencil  down ; we  should  on  either  side  get  a strip  of  cedar 
wood,  while  the  centre  would  be  a different  substance,  the 
blacklead  or  plumbago.  Parts  that  are  in  section  are 
always  thus  rigidly  defined  in  line  drawings,  and  the  lines 
are  always  <}arefully  ruled  at  equidistant  intervals  by  means 
of  the  set-square  of  45°.  The  only  exception  that  is  at  all 
tolerated  is,  that  sometimes  in  transverse  sections  of  beams 
of  timber  it  is  allowable  to  indicate  somewhat  of  the  grain 
of  the  wood,  as  in  fig.  6.  This  is  useful  as  readily  indicating 
a difference  of  material,  as  wood  from  stone  or  brick ; but 
even  when  this  is  done,  the  general  tendency  of  the  lines 
should  be  towards  an  inclination  of  45°,  in  order  that  we 
may  realise  that  it  is  a piece  of  wood  in  section  that  we  are 
looking  at,  and  not  merely  the  end  of  a beam. 

58.  Care  must  be  taken  in  drawing  section  lines  that  all 
the  lines  are  the  same  distance  apart.  We  have  already 


DRAWING  NUTS  BY  THE  SET-SQUARE, 


31 


dwelt  on  this  in  our  remarks  on  the  section  pen,  but  the 
counsel,  from  its  importance,  may  very  well  be  repeated 
here.  Tig.  7 is  a type  of  the  sort  of  thing  done  by  pupils 
whose  care  fades  away  under  a flagging  interest ; the  begiii- 
ing  is  good,  but  the  good  start  is  not  maintained  to  the 
end. 

59.  We  turn  now  to  our  other  set-square,  the  one  having 
angles  of  90°  and  60""  and  30'',  and  endeavour  to  point  out 
what  useful  service  this  can  render.  It  is  at  once  evident  that 
lines  perpendicular  to  the  T square  or  straight-edge  can  be  as 
readily  drawn  with  this  as  with  the  set-square  we  have  just 
been  describing,  as  in  each  case  one  of  the  angles  is  90°  One 
great  use  of  the  ''60''  set-square  is  in  its  application  to 
isometrical  drawing : we  may  have  occasion  to  explain  the 
nature  of  this  farther  on,  if  we  have  space  to  consider  what 
we  can  do  with  our  instruments  when  we  have  got  them ; at 
present  our  readers  must  take  our  assertion  of  the  great 
value  of  the  set-square  of  60°  in  isometrical  drawing  on  trust. 

60.  This  form  of  set-square  is  exceedingly  useful  again  in 
all  decorative  or  other  work  in  which  equilateral  triangles  or 
hexagons  enter.  The  three  angles  of  an  equilateral  or  equal- 
sided triangle  are  each  60°,  the  triangle  is  therefore  also 
equiangular  or  equal  angled.  Figs.  8,  9,  and  10  are  all  to  be 
worked  by  aid  of  the  T square  and  this  form  of  set-square : 
every  line  in  all  these  flgures  is  either  horizontal,  vertical, 
or  at  an  angle  of  either  30°  or  60° 

61.  As  nuts  in  engineering  construction  are  ordinarily 
hexagonal  in  shape,  we  give  in  fig.  ii  a ready  means  of 
drawing  them  by  means  of  this  set-square  and  a straight-edge. 
The  set-squares  are  placed  in  position  on  the  edge  of  the  ruler, 
and  it  will  readily  be  seen  how  they  may  be  slipped  along 
it  and  form,  with  the  aid  of  the  straight-edge,  every  line 
of  either  of  the  two  positions  given  of  these  hexagonal  nuts. 


32 


MA  THEM  A TICAL  INSTR  VMENTS, 


62.  As  the  great  value  of  these  two  forms  of  set-squares  is 
wholly  dependent  upon  their  being  what  draughtsmen  term 
''  true,”  we  give  in  figs.  1 2 and  1 3 simple  problems  in  geo- 
metry by  which  their  accuracy  may  be  conclusively  tested. 
Having  drawn  a line  on  the  paper  with  the  T square,  assume 
any  point  in  it,  as  A.  From  A as  centre,  with  any  radius, 
draw  an  arc  having  one  of  its  extremities,  B,  in  the  straight 
line.  From  B,  with  the  same  radius,  cut  off  a point,  C,  on  the 
arc ; and  from  C again  with  the  same  distance  cut  off  point 
D on  the  arc.  From  points  C and  D as  centres  with  the 
distance  CD  as  radius  describe  arcs  that  shall  cut  in  point 
E.  Draw  line  EA,  and  it  will  be  perpendicular  to  the  ori- 
ginal line.  This  line  EA  will  cut  the  arc  in  point  F.  From 
point  F as  centre  and  the  same  distance  as  radius  that  we 
have  been  using  all  through,  i.e.,  BC,  cut  the  arc  in  point  G. 
Draw  a line  from  A through  G and  it  will  make  with  line  AB 
an  angle  of  30°.  Draw  a line  from  A through  C and  it  will 
make  with  line  AB  an  angle  of  60°.  By  drawing  this 
geometrical  construction  to  a good  size,  say  starting  with 
a radius  AB  of  i J inches,  we  can  readily  see  how  far  our 
set-squares  are  in  accordance  with  the  true  angles.  The  T 
square  or  straight-edge  would  be  placed  to  line  AB,  and  the 
set-square  would  then  rest  on  it  and  have  its  three  angles  in 
turn  placed  at  point  A.  If  the  lines  of  the  sides  of  the  set- 
square  agree  respectively  with  AG,  AC,  and  AF,  the  instru- 
ment is  reliable ; if  they  do  not  do  so,  the  truest  economy  is 
to  discard  it,  as  it  can  only  be  a continuous  source  of  error.  . 

63.  To  test  the  set-square  of  45°  the  problem  is  commenced 
in  the  same  way,  a perpendicular  line  EA  being  drawn  to  AB. 
To  get  the  angle  of  45°  we  bisect  the  right  angle  BAF  by 
drawing  arcs  from  centres  F and  B;  these  will  intersect 
in  point  G.  It  is  immaterial  what  radius  is  employed  for 
these  arcs,  care  being  only  taken  that  the  same  radius  is  used 


11 


THE  imm 


Of  THE 

!!S!VEH£!iy  0?  !L 


PARALLEL  AND  PERPENDICULAR  LINES, 


33 


from  both  B and  F,  and  that  it  is  of  sufficient  size  to  enable 
the  arcs  to  cut  each  other  in  point  G.  It  will  be  evident 
that  unless  the  distance  taken  is  somewhat  greater  than 
half  the  distance  between  B and  F,  these  curves  would  never 
intersect  each  other  at  all.  The  line  GA  makes  with  BA  an 
angle  of  45°.  We  have  in  each  of  these  geometrical  figures 
added  the  actual  set-squares  in  a sufficiently  approximate 
position  to  the  lines  of  the  angles  to  render  the  practical  work- 
ing out  of  the  problem  quite  clear. 

64.  We  have  now  to  explain  the  ready  method  by  which, 
by  means  of  these  two  set-squares  working  together,  we  are 
enabled,  without  aid  of  any  other  ruling  instrument,  to  draw 
any  number  of  lines,  either  parallel  or  perpendicular  to  each 
other.  We  will  suppose  that,  in  the  first  place,  we  desire  to 
draw  a series  of  parallels.  If  our  readers  will  turn  to  fig.  14, 
they  will,  we  imagine,  readily  gather  the  method  employed ; 
briefly,  however,  it  is  this : — One  of  the  set-squares  in  our 
figure  marked  A is  placed  so  that  one  of  its  edges  coincides 
wdth  the  line  to  which  all  the  others  are  to  be  parallel.  Set- 
square  B is  then  placed  in  contact  with  one  of  the  other 
edges,  and  firmly  held  by  the  left  hand  while  set-square  B is 
moved  by  the  right  hand.  All  lines  drawn  by  the  edge  of 
the  set-square  which  was  used  for  the  first  line  will  be 
parallel  to  it.  It  is  not  absolutely  necessary  to  have  a 
second  set-square ; a straight-edge  or  6-inch  rule  may  supply 
the  place  of  the  second  one,  as  all  that  is  really  needed  is  a 
line  for  the  first  set-square  to  travel  upon. 

65.  To  draw  lines  perpendicular  to  others,  set-square  B (see 
fig.  15)  is  first  placed  parallel  to  one  of  the  first  series  of 
lines  by  the  method  just  explained  for  drawing  parallels ; it 
is  then  firmly  held  by  the  left  hand  while  set-square  A is 
placed  on  it  as  shown.  This  second  square  is  drawn  along 
the  first,  and  all  lines  drawn  by  its  aid  will  be  parallel  to 

c 


34 


MATHEMATICAL  INSTRUMENTS. 


each  other  and  perpendicular  to  the  first  series.  If  our 
readers  will  only  bestow  a little  care  and  give  some  little 
practice  to  it,  they  will  readily  fall  into  the  modus  ojperandi  ; 
and  having  once  learned  this  use  of  the  two  set-squares,  they 
will  find  their  knowledge  of  constant  service.  A little  more 
practice  and  a little  more  thought  bestowed  in  addition  will 
soon  enable  our  beginners  not  only  to  draw  lines  parallel  or  per- 
pendicular to  each  other,  but  also  at  30°,  45°,  or  60°  with  each 
other,  by  using  the  other  angles  of  the  set-squares.  One  great 
practical  point . to  be  observed  is,  that  the  second  set-square 
must  be  placed  gently  against  the  first ; a sudden  knock  or  jerk 
at  either  end  of  it  would  throw  it  a little  from  its  true  position, 
and  aU  the  lines  then  drawn  would  be  wrong,  thanks  to  the 
false  start.  The  thing,  we  readily  add  for  the  comfort  of  the 
novice,  is  really  much  easier  than  any  written  description  of 
it  is  able  to  convey.  Five  minutes’  observation  of  an  old 
hand  ” at  work  would  at  once  teach  the  necessary  manipu- 
lation. 

66.  For  civil  engineers  a series  of  very  similar  things 
has  been  provided  under  the  name  of  batters  and  slopes.” 
Brickwork  or  masonry  is  said  to  batter  when  the  face  of 
the  work  is  not  upright.  We  may  see  this  style  of  construc- 
tion very  well  in  deep  railway  cuttings  and  the  masonry  of 
reservoirs,  where  there  is  a heavy  pressure  either  of  earth  or 
water  to  be  resisted.  This  battering  varies,  and  its  amount 
is  expressed  in  figures.  Thus  a batter  of  i in  4 means  that  for 
every  four  feet  in  height  the  wall  is  one  foot  from  the  per- 
pendicular. If,  then,  we  cut  a piece  of  wood  into  a rectangle 
having  one  side  eight  inches  long  and  the  other  two,  and  then 
proceed  to  join  the  extremities  by  a straight  line,  this  straight 
line  will  give  the  required  angle  for  the  face  of  the  work,  a 
batter  of  i in  4.  Other  batters  are  made  of  other  useful 
angles,  as  i in  6,  i in  8,  or  i in  10 ; and  the  batter,  whatever 


SLOPES  AND  PITCHES. 


35 


it  is,  is  always  indicated.  By  the  use  of  these  much  time  is 
saved,  as  the  instrument  is  at  once  put  in  the  horizontal  line 
made  by  the  T square,  and  the  necessary  slope  is  obtained 
directly. 

67.  Slopes  are  practically  the  same  thing ; they  are  used 
for  sections  of  earthworks  in  railway  and  other  engineering 
drawings.  Where  some  recognised  slope,  such  as  2 to  i or  3 
to  I,  is  employed,  it  is  a decided  gain  to  have  an  angle  ready 
to  hand  that  at  once  gives  it.  The  ordinary  set-square  of  45° 
may  be  considered  as  a slope  showing  i to  i,  as  the  sides  of 
it  that  are  perpendicular  to  each  other  must  always  be  the 
same  size,  or  the  other  angles  would  not  be  45°.  The  slopes 
and  batters  are  ordinarily  sold  in  sets  of  half  a dozen.  They 
are  only  really  of  service  to  those  working  in  the  offices  of 
civil  engineers. 

68.  Architects  occasionally  use  what  they  term  pitches, 
set-squares  cut  to  the  various  angles  that  are  most  ordinarily 
used  in  drawing  the  pitches  or  slopes  of  roofs.  They  are 
very  similar  in  their  nature  to  the  batters  and  slopes  of  the 
engineer;  we  need,  therefore,  do  no  more  than  just  briefly 
name  them. 

69.  By  the  combined  use  of  the  T square  and  the  “45’’ 
and  60  ’’  set-squares  all  parallel  lines  can  be  so  readily  pro- 
duced that  what  is  termed  the  parallel  ruler  is  now  rarely 
employed.  It  is  occasionally  put  in  boxes  and  sets  of  instru- 
ments, but  it  is  of  very  little  real  service.  It  is  composed  of 
two  parallel  strips  of  wood  joined  together  by  metal  bars,  and 
playing  freely  on  each  other.  ‘W^hen  it  is  required  to  draw  a 
line  parallel  to  a given  line,  the  top  edge  of  the  upper  bar  is 
placed  to  the  line,  and  the  bottom  bar  drawn  down  as  far  as 
the  metal  holdings  will  allow ; the  bottom  bar  is  then  held 
firmly,  and  the  top  bar  is  brought  down  to  the  point  required 
for  the  new  line.  As  the  two  metal  pieces  are  just  the  same 


36 


MATHEMATICAL  INSTRUMENTS. 


length,  the  two  pieces  of  wood  will  always,  whether  close 
together  or  far  apart,  be  parallel  to  each  other.  The  instru- 
ment has,  however,  many  practical  defects,  which  any  one 
using  it  would  soon  detect,  but  on  which  we  need  not  here 
linger,  as  the  thing  has  grown  to  all  intents  and  purposes 
obsolete. 

70.  A decided  improvement  on  the  old  parallel  rule  is  what 
is  known  as  the  rolling  parallel.  The  body  is  composed  of 
one  long  piece  of  vulcanite,  ebony,  ivory,  electrum,  or  brass ; 
in  the  middle  of  this  a narrow  opening  is  made  that  extends 
nearly  to  each  end  of  the  ruler.  At  each  extremity  of  this 
opening  a wheel  is  placed,  and  these  wheels  are  connected 
together  by  an  axle  that  runs  from  one  to  the  other.  These 
wheels  are  slightly  grooved  on  their  edges,  so  as  to  give  them 
a better  hold  and  ''  bite  ” to  the  paper,  and  of  sufficient  dia- 
meter to  raise  the  ruler  a little  above  the  work.  A strip  of 
metal  is  put  over  the  axle ; it  allows  it  perfectly  free  play, 
and  at  the  same  time  gives  a convenient  raised  part  or  ridge 
all  down  the  centre  of  the  ruler,  by  which  it  may  be  held  and 
moved  on  the  work  as  required.  When  the  instrument  is  in 
use,  one  edge  of  it  is  tilted  lightly  down  to  touch  the  paper, 
and  the  required  line  is  then  drawn.  To  draw  others  parallel 
to  it,  the  edge  of  the  ruler  is  again  raised  from  the  paper,  the 
left  hand  is  placed  in  the  centre  of  the  raised  bridge  that 
protects  the  axle,  and  the  ruler  is  gently  moved  on  its  two 
wheels  to  any  required  distance,  its  edge  then  being  depressed 
again  and  the  new  line  drawn. 

71.  A little  practice  will  speedily  enable  any  one  to  use 
the  rolling  parallel  rapidly  and  correctly.  Care  must  be  taken 
that  the  paper  is  quite  smooth,  any  little  crumb  or  fragment 
of  india-rubber  will  suffice  to  retard  an  instant  one  of  the 
wheels,  and  then  the  direction  of  the  ruler  becomes  slightly 
changed  and  the  parallelism  of  lines  is  lost.  Where,  there- 


THE  ROLLING  PARALLEL  RULE. 


37 


fore,  a great  number  of  such  lines  requires  to  be  drawn,  it  is 
well  not  to  trust  too  entirely  to  the  accuracy  of  each  line  as 
it  follows  the  other,  but  to  occasionally  test  the  truth  by 
running  the  parallel  up  to  the  first  line  drawn  and  see  if  it 
still  agrees  with  it.  When  this  rolling  parallel  is  being  used, 
the  board  on  which  the  drawing  is  fastened  should  be  flat,  as 
the  wheels  revolve  very  easily,  and  may,  if  the  surface  be 
slanting,  speedily  deposit  the  ruler  on  the  floor.  The  price 
of  a rolling  parallel  varies  so  greatly — from  differences  in  size, 
finish,  and  material — that  w^e  can  scarcely  say  what  the  cost 
of  one  to  our  student  should  be.  On  reference  to  the  cata- 
logue of  one  of  our  best  makers,  we  see  that  a 6-inch  ebony 
may  be  got  for  a shilling,  while  a 36-inch  solid  brass,  includ- 
ing a case  to  preserve  it  from  harm,  involves  paying  loos. 
Somewhere  within  these  limits  the  novice  should,  on  due  con- 
sideration of  his  pocket  and  the  advice  of  those  competent  to 
give  it,  find  the  happy  medium.  Eeally  good  work  implies 
both  time  and  skill,  and  both  these  tend  to  make  the  result 
high  in  price ; but  in  mathematical  instruments  nothing  but 
really  good  work  is  good  enough  for  the  purpose.  A few 
really  satisfactory  instruments  are  worth  more  than  a whole 
boxful  of  inferior  things,  and  this  the  daily  working  with 
either  sort  will  very  quickly  prove. 


( 38  ^ 


CHAPTEK  IV.  . 

The  drawing-board — Sizes  to  get — Thorough  seasoning  of  the  wood — 
Trueness  of  edge — Paper  to  he  put  truly  on — Kough  board  for  cut- 
ting on — Hints  on  repinning  paper  dow'u — Overlapping  edges  of 
paper  to  be  avoided — Materials  used  for  boards — Cost  of  boards — 
Cross  strengthenings  at  back — Both  sides  of  a board  not  to  be  in 
use  at  once — Centrolinead — Its  nature  and  use — Various  forms  of 
it — Excentrolinead — Lengthening  out  lines — Drawing  lines  by  a 
chalked  cord — Line  drawings — Various  kinds  of  lines. 

72.  Though  perhaps  the  drawing-board  is  scarcely  an  item 
that  many  of  our  readers  would  include  if  asked  to  make  out 
a set  of  mathematical  instruments,  it  is,  nevertheless,  like  the 
T square,  so  useful,  indeed  so  essential,  an  accessory,  that  our 
book  would  be  practically  very  incomplete  were  we  to  fail  to 
make  some  little  mention  of  it.  A drawing-board  is  one  of 
the  first  requisites ; without  it,  very  little  indeed  can  be  done. 

73.  Even  in  so  simple  a matter  apparently  as  a piece  of 
wood  to  put  one’s  drawing  on,  many  considerations  as  to  size, 
sort  of  material,  and  therefore  price,  come  into  play.  As  to 
the  size  of  the  board,  it  is  impossible  to  say  much,  as  that 
must  altogether  depend  upon  the  nature  of  the  work  for 
which  it  will  be  used.  In  schools  of  art  throughout  the 
country,  what  is  termed  imperial  size  is  the  one  most  com- 
monly used,  because,  by  a regulation  of  the  Science  and  Art 
Department,  drawings  sent  up  to  London  for  competition  for 
medals,  &c.,  must  be  of  that  size.  On  the  other  hand,  begin- 
ners often  use  smaller  sizes  of  paper,  and  a large  board  then 


THE  DR  A WING-BOARD. 


39 


becomes  rather  a nuisance,  as  it  is  heavy  to  move  about,  and 
takes  up  much  room,  either  when  placed  against  the  wall 
or  when  in  use  on  the  table.  The  board  should  be  slightly 
larger  than  the  paper  used  with  it ; a margin  of  an  inch  or 
so  all  round  will  suffice ; but  our  readers  will  readily  see  that 
large  boards  have  this  great  advantage  over  small  ones,  that 
while  one  can  do  small  drawings  on  a large  board,  large  draw- 
ings cannot  be  done  on  a small  board. 

74.  As  a general  rule,  three  boards,  one  of  31  inches  by  22, 
another  of  22  by  16,  and  a third  of  16  by  1 1,  would  be  found 
a most  useful  equipment.  These  sizes  are  known  as  full 
imperial,  full  half-imperial,  and  full  quarter-imperial,  as  they 
just  take  either  paper  of  imperial  size,  or  the  same  paper  divided 
in  half  or  in  quarters.  The  word  full  ” means  that  there 
is  a little  margin.  An  imperial  board,  as  we  see,  is  3 1 inches 
long  and  22  inches  broad,  while  the  drawing-paper  of  that 
name  is  30  inches  by  22.  Paper,  according  to  its  size,  bears 
many  names,  such  as  royal,  imperial,  colombier,  double  ele- 
phant or  antiquarian;  what  these  sizes  are  we  shall  more 
fully  describe  in  our  remarks  on  paper  later  on,  but  we  men- 
tion them  now  because  they  give  their  names  to  the  drawing- 
boards  that  are  used  with  them.  These  are  all  called  regular 
sizes,  and  will  therefore  always  be  kept  in^  stock,  though  any 
maker  will  readily  make  any  ‘‘  irregular  ” size' at  a few  days’ 
notice. 

75.  It  is  most  essential  that  the  board,  whatever  its  mate- 
rial, should  be  of  thoroughly  seasoned  wood.  Nothing  is 
more  aggravating  to  the  temper,  and  destructive  of  good  work, 
than  a board  that  either  splits  down  the  middle,  or  so  far 
twists  that  it  is  always  in  a state  of  rattle  and  unsteadiness, 
owing  to  one  or  more  of  the  corners  being  in  the  air.  If  a 
board  is  not  perfectly  flat,  so  that  it  rests  evenly  on  a table, 
and  presents  a perfectly  even  and  uniform  surface  to  work 


40 


MA  THEM  A TICAL  INSTRUMENTS, 


on,  it  had  better  at  once  be  discarded.  As  a board  it  is  a 
failure,  but  it  may  make  very  good  firewood. 

76.  Where  the  only  sign  of  warping  is  shown  in  a splitting 
down  the  centre  of  the  board,  one  is  tempted  to  keep  the 
board  sometimes  in  service  for  awhile,  but  presently  the  fatal 
moment  surely  comes : in  taking  a measurement,  one  point  of 
the  compass  is  placed  unknowingly  right  over  the  fissure,  and 
a great  ragged  hole  in  the  paper  is  the  unhappy  result. 

77.  The  edges  of  a board  should  be  perfectly  straight,  or 
the  y square  will  not  work  truly,  and  they  should  make  a 
right  angle  with  each  other.  To  test  whether  the  board  is 
truly  rectangular  or  not,  the  T square  should  be  applied  to 
one  of  the  sides,  and  a line  drawn  across  the  board.  The 
square  should  then  be  shifted  to  the  bottom,  and  another  line 
drawn;  these  two  lines  should  be  perpendicular  to  each 
other,  and  to  test  whether  they  are  so  or  not,  the  geometrical 
method  given  in  fig.  12  should  be  employed.  At  the  point 
A,  where  these  lines  on  the  board  intersect  each  other,  draw 
an  arc,  and  make  the  construction  shown  in  fig.  12.  If  the 
arcs  that  cross  in  point  E do  not  have  their  point  of  intersec- 
tion on  the  line  drawn  on  the  board,  it  is  sufficient  proof  that 
that  line  is  not  truly  perpendicular  to  the  first  one  drawn. 

78.  We  have  in  figs.  16  and  17  drawn  two  boards.  The 
first  has  borne  one  test  of  the  y square,  the  other  has  not. 
In  this  case,  the  method  of  using  the  square  is  indicated  by 
the  dotted  lines.  It  is  evident  that  if  the  opposite  sides  of 
the  boards  were  parallel  the  lines  drawn  from  these  sides 
should  be  parallel  too.  This  does  not,  however,  test  whether 
the  sides  are  at  right  angles  to  each  other ; it  only  proves  that 
the  opposite  sides  are  true  to  each  other.  In  fig.  18  the 
parallelism  of  the  lines  ruled  is  a proof  that  the  opposite 
sides  are  parallel,  but  the  board  is  evidently  not  rectangular, 
and  only  the  geometric  method  can  satisfactorily  test  that. 


MAKING  THE  BEST  OF  A BAD  BOARD. 


41 


79.  In  some  boards,  it  will  be  found  that  the  lines  drawn 
from  two  of  its  adjacent  sides  are  true  perpendiculars,  while 
from  the  other  sides  they  are  not.  In  this  case,  instead  of  dis- 
carding the  board,  it  may  suffice  to  put  some  conspicuous  mark 
at  that  angle  which  is  true  and  reliable,  a star  or  something 
of  that  sort,  and  always  see  that  that  angle  of  the  board  is 
the  bottom  left-hand  corner  when  the  board  is  in  use.  The 
T square  can  then  be  used  fearlessly  on  those  two  sides. 
Fig.  19  is  a sufficiently  clear  illustration  of  what  we  mean, 
though  to  make  our  meaning  amply  obvious,  we  have 
exaggerated  somewhat ; for  we  imagine  that  even  the  veriest 
novice  would  hardly  buy  such  a board  as  we  have  there 
represented. 

80.  Care  should  be  exercised  in  putting  the  paper  on  the 
board.  Its  edges  should  be  parallel  with  those  of  the  board. 
This  may  readily  be  done  by  applying  the  "]"  square  to  one 
edge  of  the  paper,  the  top  or  left  hand,  and  then  gently 
pulling  the  paper  by  one  or  other  of  the  corners  until  its 
edge  coincides  with  the  line  of  the  square.  If  this  be  not 
done,  all  the  lines  drawn  on  the  paper  will  still  be  true  in 
their  relation  to  each  other,  if  the  board  and  f square  are  to 
be  relied  on,  as  we  may  see  in  fig.  20,  but  when  the  draw- 
ing comes  to  be  taken  off  the  board  it  will,  thanks  to  this 
carelessness  at  the  beginning,  look  like  fig.  21. 

81.  A board  that  is  good  for  nothing  else  is  often  service- 
able for  cutting  out  on,  for  trimming  the  edges  of  drawings,  for 
dividing  up  paper  into  halves,  and  so  on ; at  all  events,  such 
work  should  not  be  allowed  on  any  good  board.  The  board 
under  this  treatment  rapidly  becomes  dented  and  furrowed, 
and  this  departure  from  the  perfect  and  ideal  smoothness  is 
soon  felt  when  lines  have  to  be  drawn  or  inked-in.  If  nothing 
else  can  be  done,  one  side  of  the  board  must  be  looked  upon 
as  sacrificed,  and  all  cutting  done  on  that  side  only. 


42 


MATHEMATICAL  INSTRUMENTS. 


82.  When  drawings  are  elaborate,  the  paper  is  ordinarily 
strained  by  means  of  glne  or  paste.  The  way  to  do  this  we 
shall  explain  presently ; we  only  now  refer  to  it  just  to  say 
that  all  the  rough  edges  of  paper  that  result  from  this  should 
be  carefully  damped  off,  not  scraped  at  with  a knife.  Where, 
as  in  the  case  of  most  beginners,  the  work  is  merely  pinned 
on  to  the  board,  a certain  amount  of  care  must  be  exercised 
to  shift  the  positions  of  the  pieces  of  paper  occasionally. 
When  a drawing  is  once  pinned  down,  it  should  not,  if  pos- 
sible, be  disturbed,  as  it  takes  some  little  care  and  trouble  to 
see  that  the  work  is  put  square  again ; but  the  next  piece  of 
paper  should  be  a little  higher  or  lower,  or  more  to  the  left  or 
rirfit,  than  the  old  one,  or  the  board  ^ets  so  nunctured  in  a few 
places  with  pinholes  that  the  pins  presently  lose  their  hold. 

83.  When  a drawing  is  for  any  purpose  unpinned  and  then 
presently  replaced,  the  lines  afterwards  drawn  will  be  untrue 
with  those  already  on  the  paper,  unless  in  repinning  it  care 
be  taken  to  apply  the  T square  to  one  of  the  lines  (as  long  a 
one  as  practicable)  already  on  the  paper,  and  see  that  the  line 
on  the  paper  and  the  edge  of  the  square  are  coincident  before 
fastening  the  work  down. 

84.  A drawing  in  hand,  when  not  actually  being  worked  on, 
should  be  covered  up  with  a large  sheet  of  paper  to  preserve 
it  from  dust,  and  ordinarily  it  will  be  safer  to  lean  the  board, 
with  the  drawing  side  inwards,  against  the  wall,  than  to  leave 
it  on  the  table. 

85.  The  paper  should  never  be  so  large  as  to  have  an  over- 
lapping edge.  When  this  arises,  either  the  board  must  be 
exchanged  for  a larger  one,  or  a strip  must  be  cut  from  the 
paper.  Any  line  of  paper  projecting  beyond  the  plane  of  the 
board  is  a most  effectual  hindrance  to  the  true  working  of  the 
T square. 

86.  Deal  as  a material  for  boards  has  many  great  advan- 


THE'  LIBHARY 
OF  THE 

l!S51VEHS!TV  OF  IIL'KOIS 


DEAL  VERSUS  MAHOGANY. 


43 


tages,  though  there  are  undoubted  drawbacks.  The  chief  of 
these  is  its  tendency  to  warp  and  twist,  hut  various  methods 
have  been  adopted  whereby  this  may  he  more  or  less  effec- 
tually counteracted.  The  advantages  are — In  the  first  place, 
its  cheapness ; secondly,  its  lightness  ; thirdly,  its  cleanliness; 
while,  fourthly,  we  may  add  to  these  the  ease  with  which  it 
enables  the  drawing-pins  to  penetrate  or  the  paste  to  hold. 
Some  authorities  prefer  mahogany  as  a material,  but,  after  an 
experience  of  both,  we  should  ourselves  distinctly  prefer  deal. 
Mahogany  is  rather  apt  to  stain  the  back  of  the  paper  in  a very 
unsightly  way ; it  is  most  destructive  to  the  points  of  draw- 
ing-pins, and  in  pasting  paper  on  it  its  grain  is  so  close  that 
one  not  unfrequently  finds  that  the  whole  sheet  readily  comes 
up  again,  the  paste  having  been  unable  to  hold.  Anything 
like  polish  on  the  wood  is  very  undesirable,  though  in  some 
dealers’  catalogues  drawing-hoards  of  mahogany  are  put  down 
as  either  plain  or  Trench  polished,  the  latter  being  at  once 
the  less  useful  and  the  most  expensive. 

87.  Into  the  various  technical  methods  of  correcting  warp- 
ing we  need  scarcely  go,  as  it  is  really  more  a question  for  the 
manufacturer  than  the  novice.  His  best  plan  undoubtedly  is 
to  go  to  a really  good  maker,  and  then  place  himself  in  his 
hands.  Any  insight  that  we  could  give  into  clamping, 
panelling,  mortising,  or  screwing-up  would  be  of  little  avaA 
without  going  into  all  the  points  at  a greater  length  than  is 
really  desirable. 

88.  Large  sizes  in  deal  are  much  more  liable  to  twist  and 
warp  than  small  ones.  We  awhile  back  recommended  three 
sizes  as  useful  for  the  beginner  to  procure.  The  two  smaller 
sizes  might  very  well  be  the  ordinary  clamped  hoards,  and  in 
this  case  their  cost  would  he  about  eighteen  pence  and  three 
shillings.  The  largest  size  would,  if  clamped,  he  about  six 
shillings,  hut  it  would  he  very  untrustworthy;  and  for  the 


44 


MATHEMATICAL  INSTRUMENTS. 


extra  cost  of  two  shillings  a really  reliable  board  of  the  same 
size,  but  screwed  at  the  back,  might  be  supplied.  In  this 
latter  case  an  engineer’s  board  should  be  asked  for.  The 
prices  vary  somewhat  according  to  the  maker.  We  have 
bought,  we  see,  on  looking  at  our  memoranda,  boards  of  the 
same  size,  material,  and  quality  at  two  different  places ; at 
one  we  paid  three  shillings  and  twopence  for  them,  and  at 
the  other  two  shillings  and  fivepence.  These  were  22  by  1 6 
inch  boards.  - 

89.  It  is  sometimes  urged  against  the  larger  sizes  of  boards 
that  the  necessity  of  their  having  cross-pieces  screwed  on  to 
the  back  to  prevent  their  warping  limits  their  use  entirely 
to  one  side.  The  objection  is,  however,  more  fanciful  than 
real,  as  one  need  rarely  be  so  sorely  pressed  as  to  require 
both  sides  of  a board  at  once,  and  when  only  one  side  is 
wanted,  it  is  immaterial  what  hindrances  there  may  be  to  using 
the  other.  It  is  certainly  far  better  to  be  at  the  expense  of 
two  boards,  where  two  drawings  must  be  carried  on  simul- 
taneously, than  to  run  the  great  risk  of  spoiling  good  work  by 
using  both  sides  of  the  same  board.  In  addition  to  the  con- 
sideration of  the  grave  peril,  it  will  ordinarily  be  found  that 
when  two  drawings  are  advancing  together,  some  reason  exists 
why  the  draughtsman  should  be  able  to  refer  from  one  to  the 
other,  for  measurements  and  so  forth ; and  in  this  case  he  is 
far  better  able  to  consult  either  if  they  are  side  by  side  on 
different  boards  than  if  he  has  to  turn  the  board  upside  down 
on  every  occasion,  and  never,  until  the  drawings  are  finished 
and  cut  off,  have  a chance  of  really  seeing  the  two  at  once. 

90.  Other  instruments  for  the  production  of  straight  lines 
are  the  centrolinead  and  the  excentrolinead.  We  need,  how- 
ever, bestow  but  little  time  on  these,  since,  useful  as  they 
are,  they  will  never  be  found  in  the  hands  of  beginners. 
The  centrolinead  assumes  various  forms,  but  its  use  always 


CENTROLTNEAD  AND  EXCENTROLINEAD. 


45 


remains  the  same,  this  use  being  the  power  of  drawing  any 
number  of  radiating  lines  without  the  necessity  of  using  the 
actual  point  from  whence  they  spring.  It  will  be  remembered 
that  in  our  remarks  on  straight-edges  we  spoke  of  some  that 
were  used  in  large  perspective  drawings,  where  the  distance 
of  the  vanishing  point  from  the  actual  board  made  it  neces- 
sary to  employ  rulers  nine  or  ten  feet  long.  The  centrolinead 
is  intended  to  get  over  this  difficulty.  Where  one  has  abun- 
dant room  the  straight-edge  is  very  convenient,  but  where 
space  is  limited,  and  several  persons  have  to  work  in  one 
room,  it  is  difficult  to  allow  any  one  person  a stretch  of  some 
fifteen  feet,  from  point  to  point,  of  good  table-room.  In  brief, 
the  ordinary  centrolinead  is  a Y-lik^  form,  a long  arm  or 
ruler  jointed  to  two  others ; these  smaller  ones  work  on  two 
studs.  As  the  small  arms  are  working  on  these  the  longer 
one  always  travels  so  that  any  line  drawn  on  it  would  con- 
verge to  the  same  point  as  any  other.  It  is  unnecessary  to 
give  the  method  of  setting  the  instrument  for  use,  but  if  our 
readers  will  make  a Y-lik^  form  in  cardboard  and  work  it 
against  two  pins,  they  will  gain  a sufficiently  clear  idea  of  the 
nature  of  the  instrument. 

91.  Another  form  of  centrolinead  is  based  on  the  idea  that 
we  see  developed  in  the  rolling  parallel  rule.  In  the  parallel 
rule  the  two  wheels  are  exactly  the  same  size,  while  in  the 
centrolinead  one  of  the  wheels  is  removable  and  others  of 
varying  sizes  can  be  substituted,  the  effect  being  that  all 
lines  ruled  by  it  become  more  or  less  sharply  convergent  to 
an  imaginary  point,  according  to  the  diameter  of  the  wheel 
employed. 

92.  The  excentrolinead  is,  as  its  name  implies,  used  to 
draw  lines  that  are  thrown  out  of  the  general  centre.  If  the 
lines  of  the  arms  of  wheels,  for  instance,  all  radiated  from  the 
common  centre,  the  arms  would  increase  in  breadth  as  they 


46 


MATHEMATICAL  INSTRUMENTS, 


approached  the  circnmference,  but  in  practice  the  construc- 
tion is  the  reverse  of  this,  the  arms  broaden  as  they  approach 
the  boss.  The  excentrolinead  is  useful  in  the  drawing  of  a 
series  of  lines  of  this  character,  all  agreeing  in  the  measure 
of  their  excentricity. 

93.  In  making  large  diagrams  it  is  sometimes  necessary 
to  draw  very  long  straight  lines.  When  these  are  longer 
than  any  available  ruler  will  accomplish,  they  are  what  is 
termed  pieced-out.  The  method  is  as  follows  : — A line  is  first 
drawn  as  long  as  the  ruler  will  allow,  the  straight-edge  is 
then  slipped  along  the  line  till  it  overpasses  it  by  some  two- 
thirds  of  its  length,  and  the  remaining  third  of  the  ruler  is 
very  carefully  placed  to  the  piece  of  line  already  drawn. 
When  this  has  been  accurately  done,  the  new  portion  can 
then  be  added,  and  the  process  repeated  as  often  as  may  be 
necessary.  This  way,  it  will  readily  be  seen,  is  at  best  a 
makeshift ; it  is  very  tedious  in  its  application,  and  it  does 
not  make  any  provision  for  the  necessity  that  sometimes 
arises  of  joining  two  given  points,  the  extremities  of  such  a 
line.  In  the  piecing-out  process  the  line  that  starts  correctly 
from  one  point  may  be  some  inches  away  at  its  ending  from 
what  should  be  its  true  termination.  In  this  case  the  better 
method  is  to  use  a piece  of  cord.  A bradawl  or  other  con- 
venient holding  is  put  at  each  point ; a piece  of  cord  is  then 
rubbed  from  end  to  end  with  a piece  of  black,  red,  or  white 
chalk,  and  then  tightly  strained  and  tied  from  one  point  to 
the  other.  It  is  then  taken  between  the  forefinger  and  thumb 
at  about  its  centre,  drawn  out  some  four  or  six  inches,  and 
then  smartly  let  go  ; the  result  is  a beautifully  clean  and 
straight  black,  red,  or  white  line. 

94.  The  foregoing  include  all  the  leading  and  most  simple 
appliances  for  drawing  straight  lines,  and  we  must  next  pro- 
ceed to  the  consideration  of  the  means  whereby  curved  lines 


VARIOUS  LINES  EMPLOYED. 


47 


may  be  delineated.  We  would,  in  closing  this  section  of  our 
subject,  desire  to  say  something  of  the  lines  themselves. 

95.  Many  drawings  never  receive  the  aid  of  colour  at  all ; 
they  begin  and  end  with  being  line  drawings.  Every  line 
should  have  its  meaning,  and  contribute  to  the  understanding 
of  the  'work.  Nothing  that  is  essential  should  be  missed, 
nothing  that  is  redundant  should  be  tolerated.  In  such  a 
case,  it  is  evident  that  the  beginner  may  give  an  altogether 
false  impression ; the  mere  putting  in  of  a lot  of  lines  because 
a space  looks  empty  may  altogether  alter  the  character  of  the 
thing,  and  in  the  same  way  what  are  called  “ shadow ''  lines 
are  all  put  in  on  a rigid  system,  and  at  once  embellish  a 
drawing  and  explain  it,  while  an  ignorant  attempt  at  their 
use  may  lead  to  '‘confusion  worse  confounded,”  projecting 
shafts  being  shadow-lined  as  depressions,  and  round  piers 
made  to  look  like  square  ones. 

96.  Lines  receive  various  names ; where  continuous,  they 
are  called  full  lines ; where  little  spaces  are  left  at  intervals, 
they  are  called  broken  ; while  dotted  lines  are  those  where  the 
space  and  the  line  are  about  equal  in  size.  Others,  again,  are 
varied  in  different  ways  where  it  is  necessary  to  follow  out 
certain  points  through  a drawing  and  to  distinguish  them 
from  others ; thus  we  may  have  a dash  and  a dot,  or  a dash 
and  two  dots,  or  two  dashes  and  a dot  following  in  regular 
sequence.  Lines,  too,  are  spoken  of  as  fine,  medium,  or  heavy, 
according  to  their  breadth.  The  more  distant  parts  of  objects 
may  ordinarily  be  inked-in  in  finer  lines  than  the  rest.  Fig. 
22  gives  illustrations  of  various  types  of  line. 

97.  In  a geometrical  problem  the  parts  that  are  either  given 
at  the  commencement  or  that  form  a part  of  the  ultimate 
figure  would  be  full  lines,  while  the  lines  that  were  merely 
employed  in  the  working  out  would  be  either  broken  or 
dotted.  To  draw  an  equilateral  triangle  on  a given  base,  for 


48 


MATHEMATICAL  INSTRUMENTS. 


instance,  we  should  start  with  this  base  as  a full  line,  while 
the  arcs  that  give  ns  the  third  point  of  the  triangle  would  be 
dotted,  and  the  remaining  two  lines  of  the  figure  would,  when 
found,  be  put  in  as  full  or  solid  lines.  In  a perspective  draw- 
ing the  actual  object  is  put  in  in  solid  lines,  and  all  the  con- 
structive lines  in  dots.  Where  the  figure  is  complex,  all  the 
lines  that  go  to  the  formation  of  one  object  may  be  in  one 
kind  of  broken  line,  and  all  those  used  for  another  object  in 
a second  kind,  or  all  lines  going  to  vanishing  points  may  be 
of  a different  character  to  all  lines  going  to  measuring  points. 
In  any  drawing  of  machinery,  again,  the  lines  that  would  not 
be  seen  from  the  point  of  view  given  in  the  drawing  must  be 
of  a different  character  to  those  that  would  be  visible.  We 
need  not  further  multiply  examples,  for  enough  will  no  doubt 
have  been  said  to  indicate  the  importance  of  caution  and  the 
necessity  of  first  understanding  what  is  required  and  then 
doing  it,  rather  than  reversing  this  order,  as  the  manner  of 
some  is. 


( 49  > 


CHAPTEE  V. 

Instniments  for  drawing  curved  lines — Necessity  of  practice  in  freehand 
drawing — Various  sizes  of  compasses — The  hows — Spring  bows — 
Compass  joints — Fitting  pencil  to  compass — Management  of  pen 
point — The  lengthening  bar — Loose  joints — Compass  key — Double- 
jointed  compasses — Compass  points — Large  holes  at  centres  to  be 
avoided — Horn  centres — Their  cost — The  use  of  the  ink  compass — 
Circles  to  be  drawn  before  straight  lines  joining  them — Pocket 
compasses — Napier  compass — Pillar  compass — Beam  compass. 

98.  We  pass  now  to  a consideration  of  the  instruments 
employed  in  the  creation  of  curved  lines.  These  lines  are 
ninety-nine  times  out  of  a hundred  either  circles  or  arcs  of 
circles.  The  ellipse  occasionally  enters  into  drawings  of  an 
architectural  or  engineering  character.  The  arches  of  West- 
minster Bridge  are,  for  instance,  semi-ellipses,  and  we  must 
presently  enter  into  some  little  account  of  how  such  curves 
may  be  drawn.  Other  curves  of  a less  rigid  character  than 
the  ellipse  or  the  circle  may  be  drawn  by  means  of  instru- 
ments called  French  curves  (figs.  30,  31),  while  in  many 
cases  almost  all  that  can  be  done,  where  the  curves  are  very 
irregular  in  character,  is  to  pencil  them  in  as  accurately  as 
possible,  and  then  to  go  over  them  as  steadily  as  the  hand 
will  allow  with  a fine  pen,  not  a ruling-pen,  but  an  ordinary 
fine- nibbed  writing  pen.  One  sometimes  sees  students  trying 
to  draw  curves  by  hand  by  means  of  the  ruling-pen,  but  this 
instrument  scarcely  possesses  sufficient  flexibility  of  motion 
to  make  the  result  successful,  and  in  applying  it  to  a service 

D 


50 


MA  THEM  A TICAL  INSTRUMENTS, 


to  which  it  is  not  adapted,  there  is  some  considerable  risk  of 
spoiling  it,  and  so  preventing  it  from  discharging  its  legitim.ate 
work. 

99.  Tliongh  the  mathematical  draughtsman  can  carry  on 
his  work  a long  way  by  means  of  his  instruments,  the  time 
will  presently  come  when  these  alone  will  not  suffice.  The 
man  of  ruler  and  compass  sooner  or  later  comes  to  some  detail 
— a moulding  or  casting,  a capital  or  decorative  device — that 
£an  only  be  drawn  by  hand,  and  to  him,  therefore,  as  to  other 
draughtsmen,  the  necessity  of  some  knowledge  of,  and  power 
in,  freehand  drawing  comes  home.  Where  a series  of  points 
has  to  be  found  geometrically  through  which  a curve  has  to 
pass,  the  points  themselves  may  be  faultlessly  accurate  in 
position,  but  the  curve  drawn  through  them  will  be  excellent 
or  execrable  in  just  the  degree  that  the  draughtsman  has 
cultivated  or  neglected  this  practice  of  freehand  drawing.  As 
one  piece  of  practical  experience  is  worth  an  unlimited  amount 
of  speculative  surmise  and  theory,  we  recall  for  the  benefit 
of  those  who  may  be  disposed  to  undervalue  this  practice  of 
freehand  drawing,  and  regard  it  as  a thing  outside  their 
sphere,  a drawing  that  we  once  saw  in  an  architect’s  office. 
It  represented  the  front  elevation  of  some  building  of  classic 
design.  The  lines  of  the  pediment,  mouldings,  and  so  forth, 
were  all  carefully  and  neatly  given,  but  the  Corinthian  capi- 
tals, the  acanthus  leaves  feebly  drawn,  and  all  put  in  in  lines 
as  thick  again  as  the  rest  of  the  work,  spoilt  everything. 
These  capitals,  all  in  a row,  were  perhaps  a more  conspicuous 
test  and  failure  than  one  would  ordinarily  be  exposed  to ; but 
even  the  smallest  hand-drawn  curve  may  prove  a blemish  on 
an  otherwise  good  piece  of  work.  While,  therefore,  in  our 
present  pages  we  give  due  importance  to  the  various  kinds  of 
drawing  instruments,  it  must  not  be  forgotten  that  the  human 
hand  is  a drawing  instrument  too. 


VARIOUS  KINDS  OF  COMPASS, 


51 


100.  When  the  novice  opens  his  box,  the  greater  number 
of  the  brilliant  but  somewhat  mysterious-looking  forms  that 
greet  his  eyes  will  be  instruments  for  the  creation  of  curved 
lioes.  In  drawing  straight  lines,  the  ruling-pen  is  as  well 
adapted  for  those  an  inch  long  as  for  others  fifty  times  the 
length,  while  the  preliminary  pencilling  is  performed  by  the 
useful  ''  F ” or  H ''  that  does  not  figure  in  the  box  at  all.  In 
drawing  curved  lines,  however,  and  we  will  at  present  only 
deal  with  those  based  on  the  circle,  many  more  instruments 
are  used.  Though  we  may  say  roughly  that  all  such  lines  are 
drawn  by  the  compass,  we  find  in  practice  the  great  advan- 
tage of  having  various  sizes  of  compasses  to  produce  various 
sizes  of  circles,  and  in  most  boxes  three  such  sizes  will  ordi- 
narily be  found.  One  pair  will  be  about  six  inches  long  when 
lying  closed  in  the  box;  another  pair,  technically  called 

bows,”  will  be  about  four  inches  long ; while  a third,  the 

spring  bows,”  will  be  about  two  and  a half  inches  long. 
The  large  compasses  have  one  plain  point,  while  the  other 
may  be  either  like  it,  when  it  is  used  for  taking  measure- 
ments, or  this  plain  point  can  be  removed,  and  either  a piece 
in  which  a pencil  is  inserted,  or  one  for  inking-in  can  be 
substituted.  The  two  smaller  sizes  do  not  have  removable 
parts,  but  are  either  usable  for  pen  or  for  pencil  exclusively. 
We  have,  then,  the  large  compass  with  two  removable  por- 
tions, two  bow  compasses,  and  two  spring  bows,  all  employed 
for  the  pencilling  or  inking-in  of  circles  of  various  sizes.  We 
proceed  now  to  examine  these  a little  more  in  detail. 

101.  In  using  the  larger  instrument,  care  must  be  taken 
that  whatever  portion,  either  the  plain  point,  the  pencil 
point,  or  the  pen,  is  in  use,  should  be  carefully  fitted 
into  the  socket.  There  are  various  forms  of  socket  and 
means  of  accurate  attachment;  in  some  cases  the  principle 
being  that  of  a gripping  spring,  while  in  others  a screw 


52 


MATHEMATICAL  INSTRUMENTS. 


comes  down  and  presses  all  firmly  together ; but  whatever 
may  be  the  means  of  attachment,  care  must  be  exercised,  as  we 
have  said,  to  see  that  the  junction  is  complete  and  firm. 
Failing  this,  the  points  will  either  be  unsteady,  when  the  end 
of  a circle  may  decline  to  effect  a junction  with  its  beginning, 
or  else,  as  soon  as  the  instrument  is  held  over  the  paper, 
the  movable  portion  may  slip  out,  making  a dent  or  hole, 
smashing  the  pencil  point,  or  producing  a great  smear  of  ink 
on  the  work  in  hand. 

102.  When  the  piece  that  is  intended  to  hold  a pencil  is 
put  into  its  socket,  the  end  of  it  will  be  found  to  be  some- 
what shorter  than  the  other  leg,  the  plain  point,  of  the  com- 
pass. Were  it  the  same  length,  the  insertion  of  the  necessary 
pencil  would  throw  the  thing,  as  a whole,  longer  than  the 
other  leg,  an  undesirable  result.  In  fitting,  therefore,  a piece 
of  pencil  to  it,  care  must  be  taken  that  this  piece  shall  not  be 
so  long  as  to  make  much  difference  in  the  length  of  the  plain 
and  pencil  points.  A slight  margin  may  be  allowed  for  the 
wearing  down  of  the  pencil,  but  it  must,  at  all  events,  be  only 
a slight  one. 

103.  Small  pencils  are  often  sold  for  insertion  in  com- 
passes ; but  if  these  cannot  be  procured,  an  ordinary  H 
pencil  must  be  cut  down.  Soft-leaded  pencils  should  never 
be  used,  as  they  so  soon  wear  down  and  have  to  be  re-pointed, 
the  result  being  that  in  a very  brief  time  they  grow  too  short 
for  use,  and  time  is  lost  in  replacing  them. 

104.  It  may  seem  almost  unnecessary  to  say  that  a 
point  can  more  readily  be  cut  on  a pencil  six  inches  long 
than  on  one  an  inch  long,  but  one  so  often  sees  beginners 
forget  this  that  the  caution  is  not  altogether  needless. 
A boy  is  told  to  cut  a piece  off  his  pencil,  and  to  fit 
it  into  the  compass,  and  many,  if  not  watched,  will 
measure  the  length  required,  cut  it  off,  and  then  begin  to 


THE  LENGTHENING  BAR. 


S3 


sharpen  a point.  It  is  difficult  to  do  this  with  so  very  short 
a piece,  and  if  the  point,  during  the  attempt,  breaks  once  or 
twice,  the  piece  at  once  becomes  too  short  and  has  to  be  dis- 
carded. The  point  should,  first  of  all,  be  sharpened,  a piece  of 
appropriate  length  then  cut  off  the  pencil,  and  then  this  piece 
as  much  reduced  in  thickness  as  may  be  necessary  before  it 
can  be  fitted  into  the  space  prepared  for  it  in  the  compasses. 

105.  In  using  the  pen  point,  it  is  unnecessary  to  wipe  the 
back  nib  in  the  careful  way  that  we  do  in  the  ruling-pen, 
for  in  the  former  case  the  pen  does  not  come  against  the  edge 
of  any  ruler.  It  is,  however,  important  to  bear  in  mind  that 
no  pen  can  work  accurately  or  make  a satisfactory  line  unless 
the  two  nibs  are  pressed  equally  on  the  paper ; in  drawing  a 
large  circle,  therefore,  the  knee  in  the  upper  part  of  the  pen 
piece  must  be  sufficiently  bent  to  enable  the  points  to  touch 
the  paper  perpendicularly. 

106.  When  the  parts  are  held  in  position  by  a screw,  the 
student  must  be  careful  not  to  turn  it  so  far  that  it  comes 
out,  or  possibly  the  result  may  be  a long  and  fruitless  search 
for  it.  When  the  screw  is  once  lost,  the  instrument  is  of  no 
more  use  until  the  missing  part  is  replaced,  and  this  replace- 
ment means  time  and  money  loss. 

107.  In  some  boxes  a plain  bar,  socketed  at  each  end,  is 
included.  This  is  called  a lengthening  bar.  One  end  of  this, 
instead  of  the  end  of  the  pencil  or  pen  piece,  is  fitted  to  the 
compass,  and  the  pencil  or  pen  piece  is  fixed  into  the  other 
end.  By  this  means  a considerably  larger  circle  may  be 
drawn,  though  the  necessity  of  doing  so  seldom  arises  with 
beginners,  and  those  who  have  occasion  to  make  such  circles 
frequently  would  much  prefer  to  use  a beam  compass. 

108.  After  considerable  use  the  joint  at  the  head  of  the 
compass  sometimes  gets  a little  loose.  In  a really  good  work- 
ing instrument  the  joints  should  be  accurate  and  firm,  and 


54 


MATHEMATICAL  INSTRUMENTS. 


yeu  capable  of  easy  play.  When  a joint  is  too  tight  it  is 
difficult  to  take  an  exact  distance,  as  the  pressure  necessary 
to  move  it  will  often  jerk  the  points  beyond  the  required 
space.  On  the  other  hand,  too  great  freedom  of  movement 
is  at  least  as  objectionable,  as  the  compasses  will  not  then 
stay  as  set,  and  the  circle  may  be  already  drawn  in  ink  before 
we  perceive  that  the  instrument  has  played  us  false.  To 
remedy  either  of  these  faults  a small  piece  of  metal  called  a 
key  is  generally  placed  in  a box  of  instruments.  Its  ordinary 
form  is  shown  in  fig.  23.  One  end,  it  will  be  noticed,  ends 
in  a chisel-like  form ; this  is  used  to  turn  small  screws  ; the 
other  end  has  two  projecting  parts  that  fit  into  two  correspond- 
ing openings  in  the  head  of  the  compass,  and  enable  the  parts 
to  be  screwed  together  or  loosened  as  the  case  may  require. 

109.  In  cheap  sets  the  projecting  points  of  the  compass 
key  often  do  not  tally  with  the  openings  they  are  supposed 
to  fit,  and  in  this  case  aU  that  can  be  done  is  to  borrow  from 
some  one  else  whose  key  does.  We  have,  however,  ourselves 
so  often  found  that  the  possession  of  this  key  is  a temptation 
to  keep  altering  and  meddling  with  things  that  are  certainly 
not  the  better  for  such  meddling,  that  we  now  quietly  remove 
it  from  the  boxes  of  all  our  novices.  When  instruments  are 
bought  at  a good  place,  they  are  always  supplied  in  good 
working  order,  and  the  less  the  screws  and  other  parts  are 
experimented  on  the  better  for  the  things,  and  consequently 
for  the  owner.  When  he  arrives  at  years  of  discretion,” 
speaking  from  the  mathematical-instrument  point  of  view,  he 
may  be  trusted  with  all  that  is  needful. 

no.  In  the  common  kinds  of  compasses  the  plain  leg  is 
made  in  one  piece,  and  to  this  there  is  no  objection  when  no 
circles  of  large  size  are  required ; but  when  they  are,  as  in 
details  of  machinery,  the  plain  leg,  as  well  as  the  parts  that 
fit  into  the  other  leg,  should  have  a joint  in  it.  It  is  impor- 


USE  OF  THE  HORN-CENTRE. 


55 


tant  that  the  piece  bearing  the  point  that  is  stationary  in  the 
centre  should  be  perpendicular  to  the  paper,  as  well  as  the 
pen  or  pencil  points  that  are  drawing  the  circle.  Where 
this  cannot  be  done,  owing  to  the  want  of  this  knee,  it  will 
readily  be  understood  that  the  compass  legs,  being  broadly 
extended  like  a capital  A,  will  work  a large  hole  at  the 
centre  of  the  circle. 

111.  The  centre  should  be  only  just  visible  during  the 
progress  of  the  work,  and  with  a little  care  this  may  easily  be 
managed.  Some  instruments  have  triangular  points;  these 
are  objectionable,  as  the  necessary  rotation  in  making  a circle 
soon  produces  a large  round  hole.  The  better  sort  have  the 
ends  rounded  and  tapering  to  a fine  point,  while  others, 
again,  have  a groove  into  which  a common  needle  is  inserted 
and  held  in  position  by  a screw.  This  is  removable  at 
pleasure,  and  in  case  of  breakage  a new  point  can  readily  be 
obtained. 

1 1 2.  A large  hole  readily  forms  at  the  centre  when  several 
circles  have  to  be  struck  from  it,  as  in  drawing  the  lines  of 
the  teeth  of  wheels,  the  rim,  the  pitch  line,  the  boss,  and  so 
on.  This  is  more  particularly  the  case  when  the  wood  of  the 
board  is  soft  deal.  This  hole  is  not  only  unsightly  in  itself, 
but  also  sadly  injurious  to  good  results,  as  in  fine  work,  where 
there  are  many  concentric  circles,  the  eye  readily  detects 
a deviation,  and  this  deviation  must  result  unless  abso- 
lutely the  same  point  be  used  throughout.  To  obviate  the 
nuisance,  a little  contrivance  called  a horn-centre  is  often 
used  when  many  circles  have  to  be  struck  from  one  point. 
It  is  merely  a piece  of  transparent  horn  about  as  large  as  a 
sixpence,  and  having  on  its  under  surface  three  very  fine  steel 
points.  These  are  pressed  gently  into  the  paper.  As  the 
true  centre  on  the  paper  can  readily  be  distinguished  through 
the  horn,  the  compass  can  be  placed  as  accurately  on  the 


5^ 


MATHEMATICAL  INSTRUMENTS. 


horn-centre  as  on  the  paper  itself.  The  cost  of  these  little 
things  should  be  about  threepence  or  fourpence  each;  they 
are  well  worth  their  slight  cost,  and  the  little  room  they  take 
up  in  the  box. 

1 1 3.  In  drawing  circles  in  ink,  the  pen  should  be  stopped 
as  soon  as  the  circumference  is  completed,  or  very  often  the 
line  thickens  when  it  is  gone  over  again.  Sometimes,  as  in 
shadow-lining,  this  is  the  very  effect  we  want  to  produce, 
but  where  it  is  desired  to  have  the  line  of  equal  thickness,  no 
partial  re-going  over  of  any  portion  of  it  should  be  allowed. 
This  is  a point  so  readily  tested  that  our  beginners  may  with 
advantage  try  it,  the  best  way  of  impressing  the  fact  on  their 
memory. 

1 14.  Bow  compasses  are  much  like  those  we  have  already 
been  describing,  except  that  they  are  smaller  and  that  the 
points  are  not  removable.  Each  bow  is  used  for  one  special 
purpose,  either  for  inking-in  small  circles  or  for  pencilling 
them.  The  head  is  not  used  in  rotating,  as  in  the  larger 
kinds,  as  a handle  is  added  above  it.  Bow  compasses  are 
either  single  or  double  jointed.  The  former  will  make  a circle 
of  about  three  inches  in  diameter,  while  the  latter  will  pro- 
duce one  almost  double  this  size.  It  is  necessary,  as  we  have 
already  seen,  that  the  pen  should  be  as  nearly  as  possible  per- 
pendicular to  the  paper.  In  single-jointed  instruments  the 
points  soon  begin  to  lose  this  position,  and  only  a small  circle, 
therefore,  can  be  made.  On  the  other  hand,  the  double- 
jointed  bows  are  more  trouble,  as  three  joints  have  to  be 
regulated  instead  of  one, — the  joint  that  works  the  two  legs, 
and  the  smaller  joint  in  each  of  the  legs  themselves. 

1 15.  Spring  bows  are  the  smallest  form  of  compass  made. 
They  are  ordinarily  sold  in  sots  of  three,  and  are  either  placed 
in  the  general  box  or  supplied  in  a small  box  of  their  own. 
In  the  latter  case  they  should  cost  about  seven  to  nine 


THE  USE  OF  THE  SPRING  BOWS. 


S7 


shillings.  The  three  instruments  consist  of  the  dividers,  the 
ink  bows,  and  the  pencil  bows.  The  sides  are  of  steel  and 
form  two  springs ; a screw  is  fastened  to  one  of  the  legs  and 
passes  through  the  other;  a small  nut  travels  on  this,  and 
by  its  means  the  two  sides  can  be  extended  or  compressed 
until  the  points  give  the  radius  that  is  wanted.  The  dividers 
are  very  useful  in  setting  off  small  spaces  and  the  pencil  and 
ink  spring  bows  will  make  small  circles  up  to  about  an  inch 
diameter.  They  are  very  serviceable,  therefore,  in  drawing 
small  nuts,  bolt  heads,  and  the  like. 

1 1 6.  One  great  advantage  of  the  spring  bows  is  the  ease 
and  accuracy  with  which  they  may  be  set  by  means  of  the 
screw.  In  the  larger  compasses  there  is  always  a possibility 
that  the  distance  at  which  the  points  are  apart  may  be 
slightly  altered  by  some  little  knock  or  jerk.  In  the  spring 
bows  this  is  not  possible,  and  this,  where  a great  number  of 
similar  measurements  has  to  be  made,  is  a very  great  advan- 
tage. The  distance,  once  set,  remains,  and  at  the  completion 
of  the  day’s  work  they  can  be  put  in  the  box  unaltered,  and 
next  morning  or  next  year  taken  out  again  and  the  work 
accurately  resumed. 

1 1 7.  Where  a choice  is  possible,  or,  in  other  words,  nine 
hundred  and  ninety-nine  times  in  a thousand,  it  is  always 
better  to  draw  circles  or  arcs  first,  and  then  add  any  tan- 
gential straight  lines  to  them,  than  to  draw  the  straight 
lines  in  the  first  place  and  then  unite  the  curves  to  these. 
This  may  not  appear  a point  of  any  great  moment,  but  prac- 
tically it  is  found  that  it  is  very  much  easier  to  produce  good 
work  by  the  first  proceeding  than  by  the  second. 

1 1 8.  Various  modifications  of  the  preceding  types  of  com- 
passes are  made ; with  many  of  these  we  need  scarcely  trouble 
the  reader,  but  others,  such  as  the  beam  compass  or  the 
proportionah  call  for  some  little  remark.  Before,  however, 


58 


MATHEMATICAL  INSTRUMENTS. 


referring  to  these,  we  may  mention  some  few  of  the  various 
forms  of  pocket-compass. 

1 19.  Where  engineering  or  architectural  works  are  actually 
in  course  of  construction,  the  ruling  spirit,  be  he  called  civil 
engineer,  clerk  of  the  works,  or  architect,  will  often  find  the 
great  utility  of  simple  instruments  that  may  be  carried, 
without  damage  either  to  themselves  or  their  owner,  in  the 
pocket,  as  reference  can  thus,  in  case  of  any  question  arising, 
be  at  once  made  to  any  measurement  on  the  drawings. 

120.  The  most  useful  type  of  pocket  compass  is  the  hTapier. 
When  open,  this  forms  a pair  of  double-jointed  instruments, 
available  either  as  dividers  or  pen  and  pencil  points.  When 
closed,  the  joints  bend  so  far  that  these  points  are  folded 
within  the  instrument,  its  upper  part  being  hollowed  out  to 
receive  them.  It  is  then  only  about  three  inches  long,  and 
goes  readily  into  the  waistcoat  pocket.  A good  pair  of  Napier 
compasses  would  cost  about  fifteen  shillings. 

1 2 1.  The  pillar  compass,  again,  is  a very  useful  instrument. 
The  upper  half  has  its  sides  hollow,  and  into  these  the  other 
portions  may  be  slipped.  These  other  portions  are  the  small 
compasses,  pen  and  pencil,  complete  in  themselves.  When  a 
small  circle  is  required,  one  or  the  other  is  used  alone ; but 
when  a larger  pen  or  pencil  circle  is  necessary,  each  of  these 
smaller  pairs  is  stretched  open  until  they  approach  a straight 
line,  and  one  lesj  of  each  is  thrust  into  the  tubular  les^s 
of  the  larger  piece.  If  a pen-circle  be  required,  the  pencil 
compass  has  its  pencil-end  slipped  into  the  tube  and  the 
plain  end  left  out,  while  the  pen  compasses  have  their  plain 
point  thrust  into  the  tube  and  the  pen  left  out.  We  have 
now  a large  pair  of  compasses  fully  prepared  for  action.  It 
will  readily  be  understood  that  for  a pencil  circle  of  large 
size  the  operation  is  reversed,  or  both  pen  and  pencil 


BEAM  COMPASSES. 


59 


points  may  be  slipped  into  the  tube,  leaving  two  plain 
points  outside  for  use  as  a pair  of  dividers  in  taking  measure- 
ments. 

122.  The  price  of  the  pillar  compasses  is  about  the  same 
as  the  Napiers.  They  are  rather  larger,  as  the  handles  of  the 
smaller  pairs  do  not  fold  in ; but  they  are  very  complete  and 
admirable  instruments  when  well  made,  as  they  give  at  once 
the  advantages  of  large  compasses  and  small  ones  in  a very 
compact  form,  and  with  no  great  expenditure  of  trouble,  as 
the  parts  are  readily  shifted. 

123.  Beam  compasses  are  employed  when  circles  of  large 
radius  are  required.  They  are,  in  brief,  pencil  or  ink  points 
which  can  be  fitted  on  to  a bar  or  tube.  This  bar  may  be  of 
any  length ; it  is  sometimes  square  in  section  and  at  others 
round.  Sometimes  this  bar  or  beam  has  one  or  more  common 
scales  marked  on  it ; in  this  case  the  heads  of  the  pen  and 
pencil  points  have  an  opening  in  them,  through  which  the 
scale  may  be  seen.  Practically,  however,  it  is  found  to  lead 
to  more  accurate  work  to  set  off  the  required  distance  apart 
of  the  two  points  of  the  compass,  the  central  point  and  that 
which  is  to  describe  the  curve,  by  adjusting  them  to  the  re- 
quired distance  marked  off  on  a separate  ruler.  In  many 
beam  compasses,  therefore,  the  bar  is  plain.  In  using  it,  the 
beam  portion  is  horizontal,  and  the  two  beam  heads,  as  they 
are  termed,  the  portions  that  hold  the  points,  are  perpendicu- 
lar to  the  surface  of  the  paper.  The  beam  heads  slip  readily 
along  the  bar,  and  as  soon  as  the  required  radius  is  obtained, 
a screw  at  the  side  of  each  of  them  is  brought  into  use,  and 
they  are  securely  held  to  the  point  desired. 

124.  The  proportional  compass  we  merely  name  here,  as 
its  name  suggests  that  it  has  some  affinity  with  the  various 
instruments  we  have  been  describing,  but  it  is  in  reality  an 


6o 


MA  THEM  A TICAL  INSTRUMENTS. 


instrument  for  measuring  distances,  proportions,  and  areas, 
and  will  find  its  true  place  in  our  section  descriptive  of  the 
various  means  of  obtaining  the  lengths  of  lines,  &c.  Though 
called  a compass,  it  has  no  claim  to  a position  in  our  pre- 
sent section,  the  description  of  instruments  used  for  drawing 


curves. 


( 6i  ) 


CHAPTEE  VI. 

The  circle  in  mathematical  drawing — Scale  form  and  gnilloche — Ellipse 
— The  oval — Approximations  to  the  ellipse  by  means  of  arcs — 
Ellipse  drawn  by  means  of  string — By  means  of  a strip  of  paper — 
The  elliptic  trammel — Conchoidograph — Entasis  of  columns — The 
spiral  line — Methods  of  drawing  it — Railway  curves — Splines  and 
weights — French  curves — Method  of  using  them — Materials  em- 
ployed for  them — Cardboard  curves. 

125.  The  various  forms  of  compass  we  have  in  our  preced- 
ing chapter  described  will  probably,  with  the  exception  of 
the  French  curve,  be  the  only  instruments  for  drawing  curved 
lines  that  the  student  will  provide  himself  with.  The  neces- 
sities of  engineering,  geometrical,  architectural,  or  decorative 
work  bring  the  circle  into  especial  prominence,  and  it  is 
chiefly  with  instruments  that  create  the  circle  that  we,  in 
practice,  find  ourselves  dealing  with.  We  have,  therefore, 
placed  all  these  together,  and  commence  a new  chapter  in 
dealing  with  the  instruments  that  are  employed  for  drawing 
curves  that  are  non-circular.  The  circle  is  seen  in  the  forms 
of  most  wheels ; we  find  it  again  in  foiled  figures,  fig.  24,  in 
the  semicircular,  ogee,  or  segmental  arch,  the  lancet-headed 
window,  or  the  vesica,  fig.  25,  or  in  the  scale  form,  fig.  26,  and 
the  guilloche,  one  of  the  numerous  forms  of  which  is  shown 
in  fig.  27. 

126.  When  a circle  is  seen  at  an  angle  it  becomes  an 
ellipse,  or  what,  iu  popular  parlance,  is  called  an  ovaL  The 


62 


M A THEM  A TIC  A L INS  TR  UM ENTS. 


two  terms  are  not,  however,  different  names  for  the  same 
thing,  and  any  one  who  professes  to  deal  with  mathematical 
instruments  and  geometrical  figures  should  know  wherein 
the  difference  consists. 

127.  A reference  to  figs.  28  and  29  will  go  far  towards  illus- 
trating the  difference  of  form  between  an  oval  and  an  ellipse. 
An  oval,  as  we  see  in  fig.  28,  and  as  the  derivation  of  the 
word  would  suggest,  is  an  egg-shaped  figure.  One  end  is 
much  rounder  than  the  other,  and  though  one  diameter,  the 
longer  one,  cuts  the  figure  into  equal  parts  or  halves,  the 
shorter  diameter  crosses  it  some  distance  from  its  centre.  In 
an  ellipse,  on  the  other  hand,  the  two  diameters  cross  in  their 
centres,  and  divide  the  figure  into  four  similar  portions.  The 
long  diameter  of  an  ellipse  is  ordinarily  called  its  major  or 
transverse  axis.  The  short  diameter  is  termed  its  minor 
or  conjugate  axis.  When  the  two  diameters  or  axes  are 
nearly  equal  in  length  to  each  other,  the  resulting  form  ap- 
proaches a circle,  but  any  proportion  between  the  two  axes  is 
possible,  from  that  which  gives  an  almost  circular  form  to  one 
that  produces  a very  elongated  figure. 

128.  If  the  reader  will  take  a penny  or  florin  and  hold  it 
upright  between  his  fore-finger  and  thumb,  it  will  be  repre- 
sented by  a circle,  if  a view  be  made  of  it  when  in  one  posi- 
tion, and  by  a straight  line  when  held  in  another  position. 
As  the  coin  is  slowly  turned  from  one  of  these  positions  to 
the  other  it  passes  through  every  variation  of  possible  ellip- 
tical form. 

129.  Various  ways  have  been  suggested  by  which  approxi- 
mations to  the  ellipse  may  be  struck  by  compasses,  but  as 
the  true  curve  of  an  ellipse  could  never  form  an  arc  of  a 
circle  these  methods  are  all  faulty  at  bottom.  The  two  really 
practical  ways  of  constructing  the  figure  are  either  by  means 
of  a piece  of  string  or  a strip  of  paper,  and  either  of  these  are 


IV. 


ELLIPSE  DR  A WN  BY  MEANS  OF  STRING.  63 


at  once  so  efficacious  and  so  easy  in  application,  that  every 
one  who  deals  with  geometrical  constructions  at  all  should  he 
familiar  with  them. 

1 30.  Where  the  work  is  large  in  scale  the  string  method  is 
preferable : the  following  is  the  method  of  its  application. 
The  two  diameters  are  first  placed  at  right  angles  to  each 
other  at  their  centres,  and  from  either  end  of  the  short  dia- 
meter, with  half  the  major  axis  as  radius,  an  arc  is  struck 
that  cuts  this  axis  in  two  points,  the  foci  of  the  ellipse.  A 
stout  needle  is  now  driven  in  at  each  focus,  and  a third  is 
placed  at  one  end  of  the  short  diameter.  A piece  of  thin 
cord  or  strong  thread  is  then  fastened,  just  clear  of  the  paper, 
to  one  of  the  focus  needles,  passed  round  the  needle  at  the 
end  of  the  short  diameter,  and  then  again  secured  to  the 
second  focus  needle,  care  being  taken  that  the  string  or  thread 
is  tightly  stretched.  The  string  should  now  look  like  the 
capital  letter  V,  more  or  less  flattened  out  according  to  the 
proportion  of  axis  to  the  other.  The  intermediate  needle, 
that  at  the  bottom  or  point  of  the  V,  is  now  removed  and  the 
pencil  placed  there  instead.  The  string  must  be  kept  tightly 
stretched,  and  as  the  pencil  is  drawn  along  in  contact  with  it 
the  curve  of  the  ellipse  is  produced.  Care  must  be  taken  to 
keep  the  pencil  in  one  position  all  the  time  ; a nearly  upright 
one  is  the  most  effective  one  for  work.  By  this  method  an 
ellipse  may  be  readily  marked  out  of  any  size ; the  cabinet- 
maker can  find  the  true  form  of  a table  top,  or  the  gardener 
can  set  out  an  elliptical  flower  bed  fifty  feet  by  thirty  if 
necessary. 

131.  As  the  tension  on  the  string  tends  to  pull  the  focus 
needles  out  of  the  perpendicular,  unsightly  holes  may  be 
made  in  the  paper  at  these  points  unless  some  little  degree 
of  care  be  exercised.  For  small  work,  say  ellipses  up  to  a 
foot  in  length,  the  method  by  means  of  a piece  of  paper  is  on 


64 


MATHEMATICAL  INSTRUMENTS, 


some  accounts  to  be  preferred,  and  this  method  we  now  pro- 
ceed to  explain. 

132.  The  two  diameters  are  placed,  as  before,  in  their 
proper  relation  to  each  other,  but  it  is  unnecessary  to  find 
the  foci.  Any  thin  strip  of  paper  having  a cleanly  cut  edge 
is  now  taken : this  piece  of  paper  is  technically  called  a 
''  trammel ; ’’  such  a piece  may  be  seen  in  fig.  29.  Any  point, 
as  0,  is  first  marked  on  it,  then  a distance,  BG,  equal  in  length 
to  half  the  short  diameter,  and  a distance,  AO,  equal  to  half 
the  long  diameter,  is  set  off* : points  AB  are  called  travelling 
points,  and  point  C is  the  index  point.  All  is  now  ready  for 
use,  and  the  trammel  is  placed  across  the  diameters,  as  shown 
in  our  figure.  Points  A and  B are  always  placed  on  the 
lines  of  the  diameters,  and  wherever  point  G may  be,  a mark 
must  be  made.  As  many  points  as  may  be  desired  may  be 
formed,  care  being  taken  to  see  that  the  travelling  points  are 
always  on  the  two  lines.  When  these  points  are  found,  the 
curve  must  be  drawn  through  them  by  hand,  it  is,  therefore, 
poor  economy  to  shirk  the  trouble  of  finding  several  of  these 
points,  as  the  more  there  are  the  easier  it  will  afterwards  be 
to  draw  the  required  curve  through  them.  We  have  indi- 
cated this  series  of  points  in  one  quarter  of  our  ellipse  in 
fig.  29. 

133.  The  use  of  the  paper  trammel  necessitates  in  the  final 
stage  hand-drawing,  and  this  in  the  case  of  all  but  very  prac- 
tised hands  is  a great  disadvantage.  To  remedy  this,  several 
contrivances,  more  or  less  effective,  have  been  contrived,  but 
such  instruments,  from  their  costliness,  will  ordinarily  rarely 
be  found  amongst  the  paraphernalia  of  the  beginner.  On 
turning  to  the  catalogue  of  one  of  our  good  makers,  we  see 
that  one  variety  is  priced  at  seven  pounds,  while  another 
costs  ten. 

134.  The  elliptic  trammel,  based  on  the  same  principle  as 


THE  CONCHOIDOGRAPH, 


65 


that  seen  in  the  paper  strip,  is  the  simplest  and  cheapest, 
though  even  this  means  an  expenditure  bordering  on  two 
pounds.  A piece  of  brass  or  electrum  is  made  in  the  form  of 
a cross,  and  a deep  groove  runs  along  the  whole  length  of 
each  arm.  This  is  carefully  placed  on  the  paper,  so  that  its 
central  lines  coincide  with  the  diameters  pencilled  out.  In 
these  grooves  two  uprights  work  and  support  a horizontal 
bar:  this  bar  has  at  its  extremity  an  ink  or  pencil  point. 
The  uprights,  it  will  readily  be  seen,  correspond  to  our  tra- 
velling points  AB,  while  the  pen  or  pencil  point  is  the  equi- 
valent of  point  C. 

135.  Where  large  ellipses  have  to  be  made,  the  string 
method  will  be  employed ; where  a few  smaller  ones  are  occa- 
sionally required,  the  paper  edge  will  be  used ; and  where  the 
necessity  of  carefully  drawing  such  figures  frequently  recurs, 
and  the  outlay  is  no  bar,  the  metal  trammel  will  be  pro- 
cured. 

136.  Some  few  other  mathematical  instruments  for  drawing 
various  kinds  of  curves  may  be  found  in  use,  but  they  may 
ordinarily  be  regarded  as  luxuries,  as  the  necessity  for  employ- 
ing them  will  seldom  arise.  We  need  only  here  briefly  refer 
to  the  conchoidograph  and  the  helicograph. 

137.  The  conchoidograph  is  an  instrument  that,  as  its  name 
implies,  is  used  to  describe  the  curve  called  by  mathema- 
ticians the  conchoid,  though  it  may  be  employed  for  other 
curves  as  well.  The  shafts  of  Greek  columns  are  not 
straight-lined,  but  have  a gentle  curvature  outwards.  This 
curvature  or  entasis,  as  it  is  more  correctly  termed,  is  struck 
by  means  of  the  conchoidal  curve,  a curve  so  called  from  its 
resemblance  to  the  beautiful  lines  of  a bivalve  shell.  The 
amount  of  diminution  varies  from  about  one-sixth  to  one- 
fourth  of  the  diameter.  The  conchoidograph  will  draw  not 
only  the  entasis,  but  also  all  the  lines  of  the  fluting. 

E 


66 


MATHEMATICAL  INSTRUMENTS. 


138.  The  spiral  line  is  sometimes  required  in  mathema- 
tical drawing,  as  in  the  spiral  spring  of  the  engineer,  or  the 
Ionic,  Corinthian,  or  Composite  volutes  of  the  architect. 
Various  means  have  been  devised  whereby  these  lines  may 
be  readily  and  accurately  drawn,  and  instruments  having  this 
end  in  view  are  termed  helicographs. 

139.  Spiral  curves  vary  a great  deal  in  form.  In  some, 
the  lines  not  only  continuously  unwind  from  the  centre,  the 
point  technically  known  as  the  eye  of  the  spiral,  but  increase 
the  distance  they  are  apart  as  well ; while  in  others,  the  lines 
as  they  unwind  remain  at  one  distance  apart  in  the  various 
revolutions.  Any  practical  work  on  geometry  will  give  one 
or  more  ways  of  striking  spirals.  A true  spiral  is  continually 
changing  its  direction,  and  no  portion  of  it  could  be  legiti- 
mately drawn  by  means  of  arcs  of  circles ; still  several  very 
useful  approximations  to  a true  spiral  are  produced  by  this 
method,  and  practically  these  are  most  useful.  A true  spiral 
has  to  be  drawn  by  hand  through  a series  of  points  geometri- 
cally obtained,  and  is,  therefore,  at  once  correct  in  theory  and 
faulty  in  result,  while  in  the  curves  struck  by  compass  the 
result  is  only  approximate,  but  has  the  greater  neatness  of 
effect  that  instrumental  work  of  this  character  must  always 
have  on  that  produced  by  hand. 

140.  As  we  desire  to  make  our  work  as  really  useful  as  pos- 
sible, we  have  no  hesitation  in  strongly  recommending  Wynd- 
ham  Tarn’s  book  on  practical  geometry.  We  have  no  personal 
knowledge  whatever  of  this  gentleman  or  of  any  one  connected 
with  him ; we  shall  not,  therefore,  we  trust,  be  uncharitably 
misunderstood  when  we  advise  others  to  get  a book  that  we 
ourselves  value.  In  this  work  many  beautiful  forms  of  spirals 
are  given,  together  with  full  directions  for  drawing  them, 
together  with  the  practical  applications  of  the  parabola, 
hyperbola,  and  ellipse.  The  catenary  curve,  the  cycloid, 


SHIP  CURVES  AND  SPLINES, 


67 


epicycloid,  hypocycloid,  and  others,  are  all  explained,  and  the 
method  of  their  delineation  given. 

14 1.  Various  forms  of  curved  lines  are  cut  out  of  pear-tree 
wood.  Sets  of  these,  varying  in  number  from  twelve  to 
seventy-two,  are  largely  used  in  the  draughtsmen’s  offices  of 
large  shipbuilding  firms,  and  others  are  used  by  the  engineers 
engaged  in  drawing  out  plans  for  railways.  As  railway  curves 
are  almost  always  portions  of  a circle,  our  reference  to  these 
should  rather  have  been  made  in  the  previous  chapter,  but 
they  were  then  overlooked.  The  radius  the  curve  represents 
is  marked  upon  it;  thus  195  on  the  curve  means  that  it  is 
struck  from  a centre  that  distance  in  inches  away.  To  show 
the  great  use  made  of  such  curves,  we  may  mention  that 
makers  advertise  complete  sets  of  one  hundred  and  thirteen, 
ranging  from  curves  of  three  inches  to  three  hundred  inches 
radius.  Such  a set  costs  between  six  and  seven  pounds. 

142.  Both  in  the  ship  and  railway  curves  it  may  happen 
that  some  little  trouble  may  be  experienced  in  finding  just 
what  is  required,  and  they,  in  any  case,  can  only  produce 
somewhat  small  lines.  To  remedy  this,  pieces  of  wood  called 
splines  are  employed.  These  are  thin  strips  of  yew,  red  pine, 
lancewood,  or  vulcanite,  varying  in  length  from  eighteen 
inches  to  eight  feet.  In  using  them  the  required  curve  is 
first  sketched  on  the  work  in  pencil,  or  a series  of  points  in  it 
obtained ; the  spline,  from  its  flexible  nature,  is  readily  bent 
to  the  line  or  points,  and  weights  are  then  placed  at  intervals 
on  it  to  keep  it  steady  and  in  its  right  position.  The  whole 
then  forms  a ruler  along  which  the  pen  is  readily  guided. 
The  cost  of  these  splines  naturally  varies  according  to  the 
size  and  material,  but  in  any  case  it  is  not  great,  and  any  one 
having  occasion  to  draw  large  and  irregular  curves  would  find 
them  invaluable. 

143.  The  weights  used  with  the  splines  are  somewhat  more 


68 


MATHEMATICAL  INSTRUMENTS, 


expensive ; they  are  of  lead,  the  metal  being  covered  with  a 
casing  of  mahogany  or  other  wood,  both  for  the  sake  of  the 
better  appearance,  and  also  as  being  more  cleanly  in  use. 
The  ordinary  form  is  very  like  a boy’s  toy  ship  turned  upside 
down,  the  sharp  point  of  the  stem  coming  down  upon  the 
spline.  This,  backed  by  the  larger  mass  of  metal  behind, 
gives  a sufficient  hold  on  the  wood,  and  the  wedge-like  form 
of  the  part  on  the  ruler  is  the  least  possible  hindrance  to  the 
hand  when  the  line  is  being  ruled.  Care  must  of  course  be 
taken  that  the  nose  of  the  weight  does  not  advance  beyond 
the  line  of  the  ruler,  or  the  pen  will  at  that  point  meet  with 
a check. 

144.  The  appliances  known  as  French  curves  are  largely  used 
in  mathematical  drawing  work.  They  are  sometimes  called 
irregular  curves,  to  distinguish  them  from  those  that,  like  the 
railway  curves  we  have  referred  to,  are  portions  of  circles. 
French  curves  are  very  various  in  form ; a reference  to  figs.  30 
and  3 1 wdll  do  more  to  explain’  their  nature  than  any  written 
description.  They  are  from  eight  inches  to  a foot  long,  and 
vary  in  price  from  about  a shilling  to  eighteenpence.  About 
fifty  different  patterns  are  obtainable,  but  for  all  practical 
purposes  half-a-dozen  will  suffice.  As  the  lines  of  the  curves 
cross  the  grain  of  the  wood  at  all  sorts  of  angles,  these  imple- 
ments are  rather  fragile ; it  is  safer,  therefore,  to  suspend  them 
by  a nail  to  the  wall  than  to  leave  them  lying  on  the  work- 
table when  not  in  use. 

145.  In  using  the  French  curve  the  general  direction  of 
the  required  lines  must  be  first  indicated  carefully  in  pencil. 
One  of  these  curves  is  then  taken  and  applied  to  the  line 
until  a portion  of  the  pencil  line  and  the  outline  of  the  French 
curve  are  found  to  coincide,  and  this  piece  is  then  drawn  in 
by  the  aid  of  the  curved  ruler.  Another  piece  is  then 
attempted  and  found,  until  by  degrees  all  may  be  done. 


8 


HOW  TO  USE  THE  FRENCH  CURVE, 


69 


Tliough  the  continiial  re-piecing  may  appear  tiresome,  it  is 
the  only  way  to  secure  good  work,  as  ordinarily  only  a small 
portion  will  really  coincide.  Great  care  must  be  exercised  to 
maintain  the  continuous  flow  of  the  line ; nothing  like  an 
angle  or  break  at  the  joinings  should  he  visible. 

146.  French  curves  are  made  either  of  pearwood  or  vul- 
canite. We  ourselves  greatly  prefer  the  former,  for  though  it 
is  much  more  fragile,  it  possesses  certain  distinct  advantages 
over  the  other.  The  mere  difference  in  cost  is  of  no  great 
moment,  for  if  eighteenpence  will  buy  a more  serviceable 
instrument  than  a shilling  will,  it  is  a great  mistake  to  buy 
the  dear  one,  i.e.,  the  unserviceable  one. 

147.  One  great  advantage  of  the  pearwood  curve  is  its 
light  colour.  Where  many  similar  curves  have  to  be  drawn, 
as,  for  example,  the  lines  of  the  thread  of  a screw,  it  is  a great 
advantage  to  be  able,  when  the  curve  is  once  formed,  to  mark 
it  off  on  the  edge  of  the  instrument.  By  this  means  absolute 
identity  all  through  the  work  is  obtained.  Owing  to  the 
black  colour  of  the  vulcanite  composition  this  valuable  aid  to 
work  is  lost,  as  any  pencil  or  ink  marks  on  its  edge  would  be 
too  indistinct  to  be  really  serviceable.  Various  little  marks 
may  be  used,  the  beginning  and  end  of  each  portion  of  any 
particular  curve  being  distinguished  by  the  same  mark.  A 
dozen  of  these  are  represented  in  fig.  32,  and  our  readers, 
when  they  see  the  sort  of  thing  required,  will  have  no  diffi- 
culty in  devising  some  few  dozens  more.  A French  curve 
that  has  seen  good  service  will  often  be  marked  all  along  its 
edges  by  these  marks,  the  records  of  past  work. 

148.  Where  a thing  is  alike  in  curvature  on  either  side  of  a 
central  line,  the  French  curve  is  first  correctly  placed  and 
marked  for  one  side,  and  then  turned  over  and  the  marks 
carried  to  the  other  side.  By  this  means,  when  the  drawing 
is  inked-in,  the  two  sides  will  be  identical  in  character. 


70 


MATHEMATICAL  INSTRUMENTS, 


149.  As  we  have  seen  that  a curved  line  of  any  length  may 
require  the  application  of  half-a-dozen  different  portions  of 
the  French  curve  to  produce  it,  it  is  sometimes  an  advantage, 
when  such  a line  occurs  several  times  in  a drawing,  to  cut 
one’s  self  entirely  away  from  the  tedious  aid  of  this  instru- 
ment. This  may  be  done  without  imprudence  by  first  drawing 
the  required  curve  very  carefully  on  a piece  of  good  cardboard, 
and  then  with  equal  care  cutting  it  out.  Cardboard,  after 
a little  while,  owing  to  the  friction  and  pressure  of  the  ruling 
pen,  becomes  soft  and  untrue  on  its  edge,  but  this  disadvan- 
tage will  hardly  appear  before  the  temporary  curve  has 
effected  its  purpose.  If  it  gives  indication  of  wear,  all  that  is 
necessary  is  to  place  it  on  another  piece  of  card  and  mark  off 
and  cut  out  a new  line  before  the  old  one  is  destroyed. 


( 71  ) 


CHAPTEE  VII. 

Dividers — Directions  for  their  use — Stepping  out  a measurement — 
Great  accuracy  essential — Geometrical  methods  for  dividing  a line 
into  any  number  of  equal  parts — The  division  of  a circle — Geomet- 
rical figures  based  on  polygons — Accumulation  of  error  in  setting 
out  divisions — Centre  lines — Triangular  compasses — How  em- 
ployed— Methods  of  drawing  an  irregular  figure — The  pricker — 
Copying  drawings  by  its  aid. 

150.  We  proceed  now  to  consider  some  of  the  commoner 
means  employed  for  finding  points,  measuring  lines,  or  con- 
structing any  required  angles.  The  first  two  sections  run 
naturally  into  each  other,  for  though,  as  we  shall  see,  we  do 
not  always,  in  finding  points,  at  the  same  time  ascertain 
measurements,  the  one  ordinarily  follows  on  the  other. 

1 5 1.  The  instruments  known  as  dividers  are  the  simplest 
and  most  familiar  method  of  finding  any  required  points. 
These  instruments  are  found  in  almost  every  box,  though  in 
some  cases  the  compasses  take  their  place.  They  resemble 
the  compass,  except  that  in  these  both  legs  end  with  a sharp 
point;  where  these  are  not  supplied,  the  compass  has  not 
only  a pen  and  a pencil  leg,  but  also  a plain  leg,  that  can  take 
their  place,  thus  forming  a pair  of  dividers.  Where  a good 
deal  of  work  is  done,  it  is  a great  saving  of  time  to  have  a 
pair  of  dividers  as  well  as  compasses,  and  in  any  case  the  less 
the  legs  of  the  compasses  are  shifted  and  altered  the  better. 
We  ourselves  always  keep  one  compass  ready  with  the  pen- 
joint  and  another  with  the  pencil-point  in,  so  as  to  be  able  to 


72 


MATHEMATICAL  INSTRUMENTS. 


pass  at  once  from  one  to  the  other  as  the  work  demands ; and 
any  one  who  has  much  compass  work  to  do  will  soon  fall  into 
the  same  plan. 

152.  Dividers  are  chiefly  useful  when  several  similar  mea- 
surements have  to  be  made.  The  points  should  be  very  sharp, 
and  the  joint  at  the  head  of  the  instrument  should  have  that 
ready  facility  of  movement  that  may  be  termed  the  happy 
medium  between  a too  great  looseness  of  the  parts  and  an 
over-stiffness  in  the  working.  Where  the  parts  are  too  loosely 
fastened,  there  is  great  risk  that  a slight  movement,  at  the 
time  imperceptible  possibly,  may  throw  out  the  measurements, 
while  it  is  difficult  to  get  the  measurements  at  all  when  the 
parts  are  too  tightly  screwed  up.  The  instruments  decline  to 
move  without  the  exercise  of  considerable  force,  and  when 
this  force  is  applied  they  will  suddenly  open  to  a point  that 
is,  at  all  events,  not  the  point  required,  and  the  patience  of 
the  operator  undergoes  a certain  amount  of  strain. 

153.  In  using  the  instruments,  great  care  must  be  exercised 
not  to  make  unsightly  holes  and  dents  in  the  paper ; the  touch 
must  be  as  delicate  as  will  at  all  suffice  to  mark  the  point. 
A row  of  minute  holes  along  a line  may  be  almost  unnotice- 
able  at  first,  but  when  the  line  is  inked  in,  they  will  assume 
a very  undesirable  prominence.  The  dividers  should  be  held 
without  stiffness,  and,  like  all  kinds  of  compasses,  only  by  the 
head,  as  any  pressure  on  the  sides  of  the  instrument  will 
almost  certainly  affect  the  measurement. 

154.  The  required  distance  should  be  very  carefully  ob- 
tained in  the  first  place,  as  the  slightest  error  multiplies 
itself  when  this  dimension  is  ''  stepped  out,”  as  it  is  technically 
termed,  along  a line.  The  distance  that  on  the  first  measure- 
ment is  a hair’s-breadth  out,  is  two  hair’s-breadths  wrong  on 
the  second  stepping  out.  A very  slight  inaccuracy  when  set 
off  twenty  times  along  a line  will  throw  matters  out  an  inch 


DIVISIONS  OF  LINES  INTO  EQUAL  PARTS,  73 


or  so  by  the  time  the  end  of  the  line  is  reached.  This  is  often 
a source  of  great  annoyance  and  perplexity  to  beginners  in 
setting  out  polygons  in  a circle  ; all  appears  to  be  going  well, 
and  then  when  the  last  side  is  set  off,  it  is  either  too  long  or 
too  short,  and  all  has  to  be  gone  through  again.  The  student 
should  never,  in  stepping  out  such  distances,  mark  the  paper 
in  any  way  until  the  successful  arrival  at  the  end  of  the  line 
justifies  it ; if  the  last  division  is  wrong,  all  are  wrong,  and 
any  marks  made  will  only  become  sources  of  error. 

155.  Wliere  a right  line  or  circle  has  to  be  divided  out  into 
any  number  of  equal  parts,  a little  consideration  beforehand 
will  often  greatly  simplify  the  work.  When  a straight  line  has 
to  be  divided  into  an  even  number  of  parts,  such  as  2,  4,  6,  8, 
10,  the  most  accurate  method  is  to  bisect  and  re-bisect  as  far 
as  possible.  Continuous  re-bisection  will  give  us  4 or  8 parts 
without  further  labour ; but  in  the  case  of  6 or  10  we  can 
only  divide  the  line  once  by  this  method,  and  must  then  in 
each  half  place  either  3 or  5 equal  parts.  Even  then  the  gain 
is  considerable,  as  we  only  have  to  step  half  the  distance  to 
ascertain  the  right  size ; for  if  one  half  is  rightly  found,  the 
other  follows  as  a matter  of  course,  whereas  if  one  measure- 
ment is  wrong  we  find  it  out  when  half  the  line  is  reached, 
instead  of  having  to  go  from  end  to  end  before  the  error  is 
detected.  A line  divided  into  four  equal  parts  by  re-bisection 
is  shown  in  fig.  36. 

156.  Where  a straight  line  has  to  be  divided  into  any  un- 
even number  of  parts,  as  7 or  1 3,  the  geometric  method  given 
in  fig.  37  is  preferable.  A line,  AB,  is  drawn  from  one  ex- 
tremity of  the  line  to  be  divided.  This  line  may  make  any 
angle  with  the  first,  and  may  be  of  any  length.  On  this  line, 
AB,  with  any  distance,  seven  equal  parts  must  be  set  off 
with  the  dividers,  and  a line  is  drawn  from  the  last  of  these 
to  the  other  extremity  of  the  line  to  be  divided,  and  lines 


74 


MA  THEM  A TICAL  INSTRUMENTS. 


parallel  to  this  are  drawn  from  all  the  other  divisions  on 
line  AB. 

157.  It  is  of  course  a matter  of  indifference  whether  the 
parts  set  off  on  AB  reach  to  the  end  of  the  line  or  not ; all 
that  is  necessary  is  that  the  last  part,  wherever  it  falls,  should 
be  joined  with  the  extremity  of  the  line  to  be  divided.  In 
our  figure  the  seventh  point  just  falls  on  the  end  of  the  line, 
but  this  is  a mere  chance.  It  will  be  at  once  evident  that  if 
the  problem  required  the  whole  line  AB  to  be  exactly  divided 
by  spaces  that  just  took  up  its  length,  one  might  as  well,  and 
in  fact  far  better,  divide  the  other  line  at  once.  This  method 
can  be  employed  for  any  number  of  sides.  We  only  dwell  on 
it  here  because  it  is  especially  useful  for,  an  uneven  number. 

158.  In  dividing  out  a circle,  3,  4,  6,  8,  12,  16,  and  some 
higher  numbers  based  on  these,  are  very  readily  obtainable. 
Two  diameters  drawn  at  right  angles  to  each  other  will  at 
once  give  four  equal  parts ; the  bisection  of  these  will  give 
eight  and  the  re-bisection  sixteen.  The  beginner  will  soon 
find,  too,  that  the  radius  of  a circle  will  just  go  six  times  round 
its  circumference.  Mathematically  the  ratio  of  one  to  the 
other  is  not  exactly  one-sixth,  but  the  difference  is  so  slight 
that  for  all  practical  purposes  it  is  non-existent.  If,  then,  a 
diameter  be  drawn,  and  the  radius  marked  off  on  the  circum- 
ference from  each  extremity,  as  in  fig.  38,  the  circle  is  divided 
into  six  equal  parts.  If  this  distance  is  only  marked  off  from 
one  extremity  of  the  diameter,  the  circle  is,  as  in  fig.  39, 
divided  into  three  equal  parts,  while  by  drawing  two  diameters 
at  right  angles  to  each  other,  and  from  each  extremity  of  both 
then  marking  off  the  radius,  we  get  twelve  equal  parts,  as 
shown  in  fig.  40.  If  all  the  points  are  joined  successively  in 
fig.  38,  a regular  hexagon  will  be  inscribed  in  the  circle ; in 
fig.  39  the  result  will  be  an  equilateral  triangle,  and  in  fig. 
40  the  figure  will  be  a regular  dodecagon. 


Vi, 


THE  LIBHARY 
OF  THE 

njjivEBsiTY  0?  \wm 


DIVISION  OF  THE  CIRCLE  INTO  EQUAL  FARTS.  75 


159.  Figs.  41  and  42  may  be  redrawn  to  a larger  scale  by 
the  student,  as  a piece  of  useful  practice  in  finding  regular 
hexagons  and  dodecagons.  The  latter  figure,  it  will  be  seen, 
is  made  up  of  dodecagons,  hexagons,  and  tetragons  or  squares. 
Each  of  these  figures  will  be  a useful  exercise  in  neatness  and 
accuracy  of  drawing,  for,  failing  these,  the  designs  will  never 
come  out  true  to  copy. 

160.  The  use  of  the  radius  is  a great  assistance  in  setting 
out  any  multiple  of  six ; thus,  for  example,  if  we  have  a wheel 
of  forty-two  teeth,  we  should  first  divide  the  pitch  line  of  the 
wheel  into  six  equal  parts  and  then  divide  one  of  these  sixths 
into  seven.  This  would  take  some  little  time  to  do,  but  when 
once  done  the  required  measurement  could  then  readily  be 
carried  all  round  by  filling  in  the  remaining  five-sixths  with 
the  same  distance  that  was  found  to  be  right  for  the  first 
sixth. 

1 6 1.  In  the  same  way  that  we  found  in  fig.  37  a ready 

geometric  means  of  dividing  a line  into  an  uneven  number  of 
parts,  we  can  by  the  aid  of  another  geometrical  construction 
divide  the  circumference  of  any  given  circle  into  any  number 
of  equal  parts.  As  the  use  of  diameters  and  radius  will  not 
meet  every  case,  for  instance,  5 or  7,  we  give  a method  that 
is  more  especially  adapted  to  these  conditions.  We  may 
mention  that  there  are  special  methods  for  finding  almost 
any  number  of  equal  parts  on  the  circumference  of  a circle, 
one  way  being  specially  and  exclusively  for  5,  another  in 
the  same  way  limited  in  its  application  to  7.  8,  9,  10, 

and  so  on,  have  all  their  special  methods,  and  these  may 
readily  be  found  in  any  standard  work  on  geometry ; but  for 
practical  purposes,  if  the  student  knows  thoroughly  one 
general  method  by  which  any  number  of  divisions  may  be 
obtained,  he  has  knowledge  sufi&cient  for  all  ordinary  needs. 
The  necessity  for  a good  sound  knowledge  of  at  least  the 


76 


MATHEMATICAL  INSTRUMENTS, 


elements  of  geometry  is  so  essential  for  all  who  would  have 
any  dealings  with  ruler  and  compass,  that  it  seems  almost 
superfluous  to  dwell  on  it.  Those  who  have  not  this  know- 
ledge may  very  readily  acquire  it  in  its  most  useful  form  by 
the  study  of  Eawle’s  ''  Practical  Plane  Geometry,”  a manual 
almost  absurdly  cheap  in  price.  It  is  most  judiciously 
written,  for  it  gives  every  needful  problem,  while  its  pages 
are  not  cumbered  with  the  many  fancy  problems  that  some 
writers  indulge  in  and  that  have  no  practical  value.  We 
have  ourselves  long  used  it  and  commended  it  to  our  pupils, 
and  we  are  glad  to  have  this  opportunity  of  still  further 
doing  so. 

162.  The  general  method  for  dividing  a circle  into  any 
required  number  of  equal  parts  is  as  follows : — The  circle 
being  struck,  a diameter  must  be  drawn.  This  diameter  is 
divided  by  the  problem  shown  in  fig.  37  into  as  many  equal 
parts  as  divisions  are  required  on  the  circumference.  Arcs 
are  then  drawn  from  each  extremity  of  the  diameter,  and 
having  the  length  of  the  diameter  as  radius.  From  the  point 
where  these  arcs  intersect  each  other  a line  is  drawn  through 
the  second  division  on  the  diameter  and  continued  until  it 
cuts  the  circumference ; the  distance  from  this  point  to  the 
nearer  end  of  the  diameter  is  one  of  the  required  distances. 
If  this  distance  be  taken  and  set  carefully  round  the  circum- 
ference, it  will  mark  off  the  required  number  of  spaces.  This 
is  the  method  employed  for  drawing  polygons  of  any  required 
number  of  sides,  these  points  on  the  circumference  being  the 
resting-places  of  their  angles.  It  must  be  noted  carefully, 
that  whatever  the  number  of  sides  wanted,  the  line  must 
always  be  drawn  through  the  second  division  on  the  diameter. 
We  have  given  this  exceedingly  useful  problem  in  fig.  43. 

163.  Where  the  parts  are  small,  as  in  setting  out  sixty 
divisions  for  the  teeth  of  a wheel  of  eight  inches  diameter. 


DETECTION  OF  ERRORS, 


77 


the  spring-how  dividers  are  almost  indispensable.  We  have 
already  in  our  remarks  on  spring-bow  pen  and  pencil  com- 
passes explained  the  way  in  which  the  finest  divisions  and 
measurements  may  be  found  and  preserved  by  means  of  a 
screw  action,  and  this  advantage  is  equally  great  in  the  case 
of  the  dividers. 

164.  Wliere  the  circumference  of  a circle  has  to  be  divided 
into  any  number  of  equal  parts,  it  is  always  well  to  draw  a 
diameter  lightly  as  a preliminary  and  start  off  the  divisions 
from  one  extremity  of  this.  The  advantage  of  this  proceeding 
is  similar  to  that  which  we  have  already  seen  in  bisecting  a 
straight  line.  The  diameter  divides  the  circle  into  two  equal 
portions,  and  we  thus  discover,  with  half  the  labour  in  stepping 
out,  whether  our  distances  are  correct.  Where  the  required 
points  form  an  even  number,  as  6 or  10,  the  half-way  point 
should  coihcide  with  the  end  of  the  diameter,  and  if  it  does 
not,  we  at  once  see  the  necessity  of  beginning  anew  with 
a fresh  measurement.  Where  the  sides  form  an  uneven 
number,  as  5,  7,  or  9,  the  method  is  not  strictly  so  appli- 
cable, but  in  practice  it  is  nevertheless  found  to  be  an  advan- 
tage, as  the  eye  readily  detects,  when  the  half-way  distance  is 
found,  whether  the  diameter  point  bisects  it  or  not. 

165.  After  a distance  has  apparently  been  carefully  taken, 
it  may  be  found  that  the  last  point  either  slightly  exceeds  or 
falls  short  of  the  true  position.  In  this  case  the  dividers 
should  not  be  hastily  and  thoughtlessly  shifted  for  a new 
attempt,  but  it  should  be  borne  in  mind  that  this  error  has 
been  a gradually  accumulating  one,  and  has  arisen  from  a 
slight  inaccuracy  that  has  increased  with  each  measurement, 
If,  then,  in  setting  seven  equal  divisions  round  a circle  or  from 
end  to  end  of  its  diameter,  we  find  at  last  that  the  seventh 
point  overlaps  a quarter  of  an  inch,  we  must  gently  close  the 
dividers  to  what  we  judge  to  be  one-seventh  of  this  distance. 


78 


MATHEMATICAL  INSTRUMENTS. 


If  our  judgment  has  been  correct  the  second  attempt  will  be 
successful.  If,  on  the  other  hand,  our  last  point  falls  short, 
we  must  extend  the  point  of  the  dividers  to  what  we  take  to 
be  one-seventh  of  the  distance  we  have  lost. 

1 66.  A little  practice  will  readily  enable  any  one  at  the 
first  or  second  attempt  to  get  the  required  divisions  properly 
spaced  out.  We  must,  however,  impress  on  all  beginners  the 
necessity  of  absolute  exactness.  “ That’s  near  enough  ” has 
spoilt  many  a drawing,  and  has  been  as  fatal  to  good  work 
as  that  other  familiar  refuge  of  the  lazy,  '^We  can  put  it 
right  in  the  inking- in.”  A circle  that  should  have  thirty-six 
equal  divisions,  but  has  really  only  thirty-five  equal  parts  and 
another  of  about  two-thirds  the  proper  size,  may  not,  on  a 
cursory  glance,  be  noticeably  wrong,  but  on  the  comple- 
tion of  the  work  the  unlucky  tooth  or  cog  that  is  only  two- 
thirds  as  big  as  any  of  the  others  beco^nes  wofully  conspi- 
cuous. 

167.  In  measuring  off  work  from  a piece  of  machinery  or 
some  architectural  construction,  or  when  copying  another 
drawing,  centre  lines  should  as  much  as  possible  be  found 
and  worked  from.  Most  details  of  engineering  or  architec- 
tural works  have  some  considerable  degree  of  symmetry  in 
their  arrangements ; centre  lines  of  shafts  can  be  marked, 
axes  of  columns  found,  and  the  like,  and  the  work  thus 
methodically  arranged  and  set  out  in  the  drawing  has  far  less 
chance  of  error  creeping  in  than  where  this  basis  is  not 
employed. 

It  is  often  an  advantage  to  be  able  to  determine  readily  the 
relation  of  three  points  to  each  other.  By  the  use  of  the 
common  dividers  w^e  ascertain  the  position  of  a second  point 
in  relation  to  the  first,  and  by  the  use  of  what  are  termed 
triangular  compasses  we  find  the  true  relation  of  one  point  to 
two  others.  This  instrument  is  especially  useful  in  land  sur- 


METHODS  OF  COPYING  IRREGULAR  FIGURES.  79 


veys,  as  the  plots  are  frequently  very  irregular  in  form.  We 
have  in  fig.  44  a representation  of  one  of  these  plots. 

169.  In  making  a copy  of  this  irregular  figure  we  might 
employ  either  the  dividers,  the  common  compass,  or  the 
triangular  compass,  and  the  method  of  use  in  each  of  these 
cases  we  will  now  proceed  to  give.  It  is  understood  that  the 
figure  is  in  each  case  to  he  the  size  of  the  original. 

170.  The  readiest  way  of  drawing  the  plot  ABCDEF 
by  the  aid  of  the  dividers  would  be  to  fix,  in  the  first  place, 
on  any  one  of  the  lines,  as  AB,  as  a base,  and  to  draw  a line 
AY  perpendicular  to  it.  Lines  parallel  to  AB  should  then 
be  drawn  from  all  the  other  points  until  they  touch  line 
AV ; these  lines  will  cut  in  points  KLMV.  The  distances 
VF,  MD,  LF,  and  KC  can  then  readily  be  obtained,  and 
the  true  position  of  the  angles  of  the  field  determined. 

1 7 1.  To  draw  this  figure  by  means  of  the  ordinary  compass 
we  take  any  one  line,  as  AB  again,  as  a base  to  work  from, 
and  having  drawn  such  a line  on  our  paper,  we  measure  the 
distance  from  A to  D,  and  draw  a line  having  that  radius 
from  centre  H.  We  then  measure  the  distance  from  B to 
point  D again,  and  with  this  radius,  and  point  B as  centre, 
we  draw  another  arc  cutting  the  first ; the  point  of  intersec- 
tion will  give  us  point  D on  our  drawing.  From  either  lines 
AB  or  AD  we  may,  in  the  same  way,  determine  point  E by 
finding  intersecting  arcs,  and  either  lines  AB,  AD,  or  DE 
will  give  us  a base  line  from  which  we  find  point  F. 

172.  The  point  to  be  found  should  lie  well  between  the 
two  points  of  the  base  from  which  the  arcs  are  struck,  or  it 
becomes  difficult  to  tell  the  exact  point  at  which  the  arcs 
really  intersect.  In  copying  our  figure  our  readers  would 
soon  detect  this,  for  they  would  perceive  that  while  the  inter- 
section of  the  arcs  at  point  C was  very  clear  when  they  were 
struck  from  points  E and  F,  it  would  be  very  difficult  to 


8o 


MATHEMATICAL  INSTRUMENTS, 


determine  the  true  point,  owing  to  the  great  similarity  of 
direction,  if  the  arcs  to  find  angle  C had  been  struck  from 
points  D and  E.  The  same  disadvantage  would  be  felt  in  a 
minor  degree  if  we  took  points  A and  B to  find  G by ; A and 
D would  be  much  better. 

173.  The  principle  of  finding  the  position  of  a third  point 
when  we  know  the  position  of  two  others,  can,  it  will  be 
seen,  readily  be  determined  by  this  method  of  intersection  of 
arcs,  but  the  triangular  compass  arrives  at  the  result  much 
more  quickly.  To  find  point  J)  by  the  intersection  of  arcs, 
we  (i)  draAV  a line,  AB,  and  (2)  measure  its  correct  length: 
we  then  (3)  from  A determine  distance  AD  with  compass, 
and  (4)  draw  this  arc  on  our  paper ; we  next  (5)  find  dis- 
tance BD,  and  also  (6)  draw  this  on  our  paper.  By  using 
the  triangular  compasses  we  arrive  at  the  same  end  as  follows : 
We  draw  (i)  an  indefinite  line  on  our  paper,  we  then  (2) 
place  the  compasses  so  that  the  three  points  rest  on  AB  and 
D in  the  example  we  are  copying ; and  we,  in  conclusion  (3) 
place  two  of  these  compass  points  on  our  indefinite  line,  their 
resting-places  giving  us  at  once  the  positions  of  A and  B, 
and  the  third  point  at  the  same  time  marking  the  position  of 
point  D.  We  in  the  same  way  triangulate  the  rest  of  the 
figure.  One  of  the  collateral  advantages  of  the  use  of  the 
triangular  compass  is,  that  the  point  is  at  once  obtained,  a 
slight  dent  or  a pencil  point  being  made  to  mark  it,  while  in 
the  other  case  a good  deal  of  pencil  work  has  to  be  rubbed 
out  when  the  arcs  have  been  used. 

174.  In  our  remarks  on  the  use  of  the  protractor  we  shall 
indicate  a method  by  which,  by  the  aid  of  that  instrument, 
any  irregular  spaces,  like  that  shown  in  fig.  44,  can  be  either 
enlarged  or  reduced. 

175.  Where  drawings  of  a bold  and  simple  character  have 
to  be  copied,  the  use  of  the  pricker  is  at  times  an  advantage. 


COPYING  DRAWINGS  BY  PRICKING. 


8i 


All  that  is  really  required  is  a fine  needle,  but  as  this  by  itself 
is  somewhat  difficult  to  hold  and  to  exert  pressure  on,  this 
needle-point  is  either  fitted  to  a special  handle  of  its  own,  or 
the  top  part  of  the  handle  of  the  ruling  pen  unscrews  and  is 
used  for  it.  Where  much  pricking  through  has  to  be  done, 
it  is  as  well  to  have  this  instrument  complete  in  itself,  and 
not  part  of  any  other.  A pricker  can  be  bought  for  a shilling, 
or  a very  good  home-made  one  can  be  made  by  means  of  a 
needle,  a penholder,  and  sufficient  strong  thread  to  bind  one 
securely  to  the  other. 

176.  Several  copies  can  be  made  at  once  by  placing  the 
requisite  number  of  sheets  of  paper  beneath  the  example, 
and  pricking  the  points  at  once  through  all  of  them.  This 
is  a very  speedy  way  of  reproducing  simple  working  drawings, 
or  making  copies  of  irregular  forms  without  need  of  either 
intersecting  arcs  or  triangular  compasses. 

177.  The  practical  points  to  be  observed  in  the  use  of  the 
pricker  may  be  readily  summed  up.  In  the  first  place,  the 
process  should  hardly  be  applied  to  an  original  drawing  that 
is  so  far  valued  that  a number  of  little  holes  all  over  it  are 
an  objection.  In  pricking  through,  great  care  must  be  taken 
that  the  original  and  the  paper  beneath  do  not  shift  on  each 
other : it  will  be  very  difficult  to  get  them  right  again,  and, 
if  they  are  not  got  right,  all  the  work  afterwards  done  is  wrong 
in  relation  to  what  has  gone  before.  The  two  papers  should 
either  be  pinned  down,  or  a weight  placed  on  some  part  not 
being  worked  at.  Where  the  drawing  is  at  all  complicated 
and  the  transferred  points  numerous,  one  corner  of  the  ori- 
ginal should  be  from  time  to  time  sufficiently  raised  to  allow 
some  number  or  arbitrary  device  to  be  placed  on  some  of  the 
leading  points.  There  is  a certain  danger  of  the  papers  shift- 
ing in  doing  this,  if  a considerable  degree  of  care  be  not  taken ; 
but.  if  it  be  not  done,  the  student  will  find,  after  he  has  been 


82 


MATHEMATICAL  INSTRUMENTS, 


laboriously  pricking  away  for  some  time,  that,  when  he  at  last 
removes  the  original  and  looks  at  his  paper,  it  presents  to  his 
dismayed  eyes  some  dozens  of  holes  scattered  over  the  sheets  : 
suggestions  of  which  it  is  hopeless  to  make  use,  as  it  is  im- 
possible to  see,  amidst  them  all,  what  points  have  to  be 
joined.  This  is  no  fancy  sketch  or  imagined  possibility : we 
have  ourselves  seen  this  look  of  dismay,  and  ''  what  has 
been  may  be.” 


( 83  ) 


CHAPTER  VIII. 

Scales,  tlieir  nature  and  construction — The  representative  fraction — 
Eeading  to  edge — Duodecimal  scales  in  common  use — Diagonal 
scales,  their  construction  and  use — Decimals — Other  scales  found 
on  protractors,  &c. 

178.  Vaeious  scales  will  always  form  a part  of  the  equipment 
of  the  draughtsman.  Drawings  made  to  scale,  as  it  is  termed, 
are  representations  of  some  object,  and  bear  some  fixed  rela- 
tionship of  size  to  it,  so  that,  when  we  once  know  what  this 
scale  is,  we  are  able  to  find  out  the  real  size  of  the  object 
that  is  represented.  To  this  end  the  scale  of  the  drawing  is 
either  named  on  it  or  actually  drawn,  the  latter  being  far 
preferable.  Ordinarily  drawings  are  made  of  less  dimensions 
than  the  constructions  or  forms  they  represent,  but  in  some 
cases,  as  in  drawings  of  microscopic  preparations,  the  reverse 
is  the  case. 

1 79.  If  a drawing  be  made  to  a scale  of  six  inches  to  the 
foot,  it  is  at  once  seen  that  its  parts  are  all  half  the  size  of 
the  corresponding  parts  in  the  real  thing:  such  a drawing 
would  be  said  to  be  half  scale.  In  the  same  way,  if  a plan 
be  drawn  to  the  scale  of  an  inch  to  the  mile,  we  see  that  a 
distance  that  really  measures  four  inches  and  a half  repre- 
sents a distance  of  four  miles  and  a half. 

1 80.  Sometimes  the  scale  is  represented  on  a drawing  by 
what  is  termed  its  representative  fraction.  This  indicates 
what  proportion  the  drawing  and  the  original  bear  to  each 


84 


MA  THEM  A TICAL  INSTRUMENTS. 


other;  thus  a drawing  half  the  size  of  the  object  would  be 
marked  as  one  foot  in  the  drawing  is  the  representative 
of  two  feet  in  the  real  thing.  A drawing  of  a yard  to  the 
mile  would  have  ttVo  its  representative  fraction,  because 
what  is  represented  by  one  yard  is,  in  the  real  thing,  a dis- 
tance of  one  mile,  or  seventeen  hundred  and  sixty  yards. 

181.  The  scales  that  are  more  commonly  used  in  land  sur- 
veying, mechanical,  or  architectural  work  are  drawn  upon 
rules  of  boxwood  or  ivory,  and  placed  in  the  instrument  box. 
By  means  of  these  much  time  is  saved,  as  the  draughtsman 
does  not  need  to  draw  the  scale,  nor  take  his  dimensions  by 
compass  from  it,  but  can  at  once,  when  the  divisions  read  to 
the  edge  of  the  rule,  place  his  scale  to  the  drawing,  and  tick 
off  the  required  distance  at  once  on  any  line  by  means  of  the 
pencil.  In  the  same  way,  when  it  is  desired  to  know  what 
the  real  size  of  any  detail  in  a drawing  may  be,  the  edge  of 
the  scale  is  at  once  applied  to  it,  and  its  dimensions  read  off. 
The  scales  we  most  often  find  on  these  rules  are  the 

h B -16,  f,  f,  ij,  i i if,  li  and  3 inch.  These  are,  in 
each  case,  divided  duodecimally ; i.e.,  the  last  division  to  the 
left  is  divided  into  twelve  equal  parts  for  inches.  A drawing, 
therefore,  of  |ths  of  an  inch  to  the  foot  would  have  a real 
distance  of  2 feet  5 inches  represented  by  a line  that 
was  as  long  as  two  spaces  of  |ths  of  an  inch  each  plus  five  of 
the  twelve  parts  into  which  one  of  the  |-inch  spaces  was 
divided. 

182.  To  set  out  a scale,  we  must  know  what  relation  in 
size  the  original  thing  and  the  drawing  are  to  bear  to  each 
other.  We  will  assume  that,  for  some  reason,  none  of  the 
scales  named  above  are  applicable,  and  that  it  becomes  neces- 
sary to  draw  one  for  ourselves,  and  we  decide  that  our  draw- 
ing shall  be  in  the  proportion  of  three  and  a half  inches  to  the 
foot.  To  make  this  scale,  we  commence  by  drawing  a line 


READING  TO  THE  EDGE  IN  SCALES, 


85 


and  marking  off  on  it  a series  of  spaces,  each  three  inches  and 
a half  long.  Each  of  these  spaces  we  reckon  as  one  foot. 
We  now  take  the  left-hand  division  and  divide  it,  by  the 
method  shown  in  fig.  37,  into  twelve  equal  parts  for  inches. 
We  then  call  the  inner  end  of  this  last  space  o,  and  from 
this  zero  we  number  off  our  feet  consecutively  to  the  right 
and  our  inches  to  the  left,  so  that  the  numbers  from  the  left- 
hand  end  of  the  scale  would  run  as  follows  : — 12,  o,  i,  2,  3,  &c. 
If  now  we  want  to  take  a measurement  of  3 feet  7 inches,  we 
place  one  end  of  the  compasses  at  figure  3,  and  extend  the 
other  leg  until  the  point  has  travelled  seven-twelfths  of  the 
distance  beyond  zero ; or  if  the  scale  reads  to  the  edge,  we  at 
once  tick  off  a distance  that  corresponds  with  these  points  in 
the  scale. 

183.  As  some  of  our  readers  may  scarcely  comprehend 
what  we  mean  by  reading  to  the  edge,’'  we  give  in  fig.  45 
an  example  of  a scale  that  does  so  read.  A very  slight 
inspection  of  it  will  show  that  all  the  divisions  are  carried  to 
the  edge  of  the  rule,  and  that  any  distance  by  it  could  at 
once  be  marked  off.  In  contrast  to  this,  our  readers  will  see 
in  figs.  33  and  34  examples  of  scales  which  do  not  read  to 
the  edge.  The  chief  use  of  this  instrument  is  to  find  angles 
by,  as  we  shall  see  presently,  and  these  scales  are  merely 
placed  in  the  parts  that  would  otherwise  be  left  blank,  an 
arrangement  that  is  at  least  far  more  serviceable  than  bare 
wood  or  ivory  would  be.  The  scales  in  fig.  33  are  the  inch, 
the  three-quarters,  the  half,  and  the  quarter-inch,  and  it  will 
readily  be  seen  that  to  transfer  any  of  these  to  the  drawing, 
it  would  be  necessary  either  to  mark  them  off  carefully  on  a 
straight-edged  strip  of  paper,  or  take  them  off  with  dividers. 
The  scales  could  not,  as  in  fig.  45,  be  directly  applied  to  the 
drawing.  The  scales  on  fig.  34  are  decimally  divided,  the 
divisions  being  45,  40,  35,  30,  25,  and  20  to  the  inch  respec- 


86 


MATHEMATICAL  INSTRUMENTS. 


tively.  If  we  want  to  find  a distance  of  65  feet  on  a plan 
drawn  to  20  feet  to  the  inch,  we  place  one  leg  of  the  com- 
pass on  figure  6 and  the  other  on  the  fifth  division  to  the 
left  of  zero,  because  each  of  the  divisions  to  the  right  of  zero 
represents  spaces  of  ten  feet,  while  the  space  to  the  left  of 
zero  is  similar  to  these,  but  has  the  single  feet  marked  on  it. 

184.  The  line  of  chords  as  the  part  marked  ‘‘Cho”  in  fig. 
33,  or  “C”  in  fig.  34,  is  called,  is  generally  put  on  the  scales 
supplied,  but  it  is  very  rarely  used,  as  whatever  it  can  do  for 
the  draughtsman  the  protractor  will  do  better  and  more 
readily.  Like  the  latter,  it  is  used  for  finding  angles:  its 
method  of  use  will  be  found  farther  on,  when  we  consider  the 
application  of  the  protractor,  sector,  and  other  angle-measur- 
ing instruments. 

185.  The  lower  part  of  fig.  34  is  taken  up  by  what  is 
called  a diagonal  scale.  As  the  principle  of  the  thing  is 
very  good,  we  will  describe  the  method  of  drawing  one.  A 
convenient  unit  of  measurement  is  taken  and  marked  off  on 
a line,  and  numbered  from  zero  to  the  right,  and  one  space  of 
equal  length  marked  off  to  the  left  of  zero,  as  in  making  a 
plain  scale.  In  a plain  scale  we  get  two  dimensions,  as  feet 
and  inches,  but  in  a diagonal  scale  we  get  three,  as  yards, 
feet,  and  inches,  miles,  furlongs,  and  chains,  or  units,  tenths, 
and  hundredths.  The  scale  represented  on  fig.  34  is  of  the 
latter  description ; there  are,  in  fact,  two  on  the  rule,  that  to 
the  left  being  a quarter-inch,  and  that  to  the  right  a half-inch 
scale  : the  figures  of  the  former  read  along  the  bottom  line, 
and  of  the  latter  along  the  top  line.  This  is  the  actual  re- 
presentation of  the  ordinary  form  of  ruler  supplied  in  instru- 
ment boxes;  but  in  fig.  35  we  have  an  enlarged  rendering  of 
a diagonal  scale  adapted  for  yards,  feet,  and  inches,  and  in  fig. 
46  another  for  reading  units,  tenths,  and  hundredths,  the 
larger  size  being  clearer  for  the  purpose  of  explanation.  In 


CONSTRUCTION  OF  THE  DIAGONAL  SCALE,  87 


setting  out  a diagonal  scale,  the  units  to  the  right  of  zero  are 
the  largest  denomination,  and  those  to  the  left  of  zero  are  the 
next  largest,  while  the  third  denomination  is  set  off  down  a 
line  at  a right  angle  to  this. 

186.  If  the  student  will  refer  to  lig.  35  he  will  see  six 
divisions  set  off  to  the  right ; these  are  yards.  The  left-hand 
division  is  divided  into  three  parts,  because  there  are  three 
feet  in  a yard.  A line  of  any  convenient  length  is  then 
drawn  at  a right  angle  to  the  first,  and  on  this  twelve 
equal  divisions  are  marked,  because  there  are  twelve  inches 
in  every  foot.  Had  the  divisions  been  representative  of 
miles,  furlongs,  and  chains,  the  unit  to  the  left  of  zero 
would  have  been  divided  into  eight,  because  there  are  eight 
furlongs  in  a mile,  and  the  line  perpendicular  to  it  into 
ten,  because  there  are  ten  chains  in  a furlong.  Lines  parallel 
to  the  first  are  now  drawn,  as  shown  in  fig.  35,  from  all  the 
divisions  that  mark  inches,  and  lines  perpendicular  to  these 
from  all  the  divisions  that  mark  yards.  The  divisions  that 
indicate  the  feet  are  marked  on  the  top  and  the  bottom 
line  in  the  left-hand  space,  and  joined  by  lines  that  are  each 
one  division  out  of  the  perpendicular,  i being  joined  to  o,  2 
to  I,  and  3 to  2.  It  is  this  portion  of  the  construction  that 
has  earned  it  the  name  of  the  diagonal  scale,  though  the 
oblique-lined  scale  would  have  been  a preferable  name,  as 
these  slanting  lines  are  not  diagonals  at  all. 

187.  Having  now  made  our  scale,  we  must  learn  how  to 
use  it.  As  all  the  horizontal  lines  are  the  same  distance  apart, 
the  spaces  between  o,  o,  and  the  first  oblique  line  o,  i,  will 
diminish  in  a regular  proportion,  and  if  the  space  o,  i,  on  the 
top  line  represents  one  foot  or  twelve  inches,  the  space  cut 
off  on  the  next  line  will  be  a twelfth  less,  or  eleven  inches, 
and  so  on,  each  diminishing  space  being  one  inch  less.  The 
spaces  cut  off  on  the  horizontal  lines  by  the  slanting  lines 


88 


MATHEMATICAL  INSTRUMENTS, 


I,  o,  and  2,  I,  and  3,  2,  are  all  equal  to  each  other,  and  in 
every  case  represent  feet.  If,  then,  we  want  to  take  a distance 
of  5 yards  i foot  6 inches  from  the  scale,  we  place  one 
leg  of  the  compass  at  A and  the  other  at  B.  This  will  give 
us  a length  of  five  yards  and  one  foot  and  six-twelfths,  or 
inches,  of  another  foot.  All  the  yard  and  foot  spaces  are 
equal  on  whichever  line  we  measure  them  ; we  therefore  only 
need  to  count  up  w^hatever  the  number  of  inches  may  be,  so 
that  in  this  case  we  measure  on  the  sixth  line  from  the 
bottom.  To  give  one  other  example,  we  will  suppose  that 
we  require  a distance  of  3 yards  2 feet  9 inches.  In  this 
case  we  should  count  nine  lines  up  the  yard  line  marked 
3,  and  extend  the  compasses  to  G,  thus  getting  at  one  opera- 
tion three  complete  spaces  of  yards,  two  of  feet,  and  nine- 
twelfths  of  the  remaining  foot,  a distance  indicated  in  our 
figure  by  CD. 

188.  If  we  have  succeeded  in  explaining  with  sufficient 
clearness  the  principle  of  construction  and  method  of  use  of 
the  scale,  fig.  35,  of  yards,  feet,  and  inches,  the  student  will 
have  little  difficulty  in  comprehending  the  diagonal  scale  we 
have  represented  in  fig.  46,  by  means  of  which  tenths  and 
hundredths  may  be  found.  In  almost  all  geometrical  exami- 
nation questions,  as,  for  instance,  those  for  candidates  for 
commissions  in  the  Engineers,  Artillery,  or  the  Line,  the 
dimensions  are  given  in  decimals.  If  the  candidate  has  to 
construct  some  figure  on  a line  3.83  inches  long,  and  cannot 
draw  the  line,  all  the  superstructure  that  he  would  have 
reared  upon  it  falls  to  the  ground. 

189.  On  the  protractor,  fig.  34,  the  diagonal  scales  bear 
the  proportion  of  200  and  400  parts  to  the  inch  respectively, 
as  in  one  case  100  parts  are  represented  by  half  an  inch,  and 
in  the  other  case  the  100  parts  are  represented  by  a quarter 
of  an  inch.  In  our  illustration  these  distances  are  greatly 


45. 


THE  LIBHARY 
OF  THE 

l!H!VERS!TY  C?  11L'K3!S 


CONSTRUCTION  OF  THE  DIAGONAL  SCALE,  89 


reduced,  but  it  will  be  readily  seen  that  the  finely  divided 
portion  at  one  end  is  as  large  again  as  that  at  the  other, 
and  if  our  readers  will  examine  the  six-inch  protractor  in 
their  instrument  case,  they  will  find  that  these  distances 
are  really  quarter  and  half-inch  spaces. 

190.  Though  distances  in  the  questions  in  examinations 
are  ordinarily  given  in  inches,  tenths,  and  hundredths  of 
inches,  and  could  be  at  once  taken  off  by  means  of  a scale  of 
100  parts  to  the  inch,  these  reduced  scales  of  200  and  400 
to  the  inch  are  very  readily  employed.  If  we  have  a scale 
of  100  to  the  inch,  and  want  any  given  distance,  as  2.13 
inches,  we  take  it  at  once  from  it ; but  if  we  have  only  the 
200  to  the  inch  scale,  we  take  the  distance  2.13  on  that  and 
double  it ; or  if  we  use  the  400  to  the  inch  scale,  we  take  the 
measurement  by  that  and  then  set  it  off  four  times. 

1 91.  The  scales  of  200,  and  especially  of  400,  to  the  inch 
have  the  further  advantage  that  by  their  aid  we  are  readily 
enabled  to  set  off  lengths  of  greater  magnitude  than  any 
given  on  the  scale.  If,  for  example,  we  want  to  find  a line 
that  shall  really  be  18.43  inches  long,  we  find  the  represen- 
tative of  that  distance  on  the  400  scale,  and  then  step  it 
along  the  line  four  times  to  obtain  the  result.  Such  measure- 
ments should  be  very  carefully  taken,  as  any  error  either  of 
shortcoming  or  excess  repeats  itself  fourfold. 

192.  To  construct  a diagonal  scale  that  shall  be  available 
for  measuring  inches,  tenths,  and  hundredths,  we  first  draw 
a line  and  mark  off  on  it  a series  of  spaces  as  ABO,  making 
them  each  one  inch  long,  if  we  propose  to  have  a scale  of  100 
divisions  to  the  inch,  or  half  an  inch  long  if  the  scale  is  to 
200  divisions  to  the  inch,  or  a quarter  of  an  inch  long  if  400 
divisions  to  the  inch  are  required.  The  left-hand  division, 
AB,  we  now  divide  into  ten  equal  parts,  and  we  draw 
another  line,  AD,  at  right  angles  to  this.  Line  AD  may 


90 


MATHEMATICAL  INSTRUMENTS. 


be  of  any  length  we  choose,  but  whatever  length  we  make  it, 
we  must  afterwards  divide  it  into  ten  equal  parts.  Prac- 
tically, therefore,  the  better  plan  is  to  draw  the  line  inde- 
finitely, and  then  commence  setting  off  from  A ten  equal 
parts,  afterwards  rubbing  out  any  part  of  the  line  that  may 
be  over.  From  all  these  divisions  we  draw  lines  parallel  to 
ABC.  The  points  on  AB  are  now  to  be  transferred  on  to 
DE,  and  the  two  sets  of  points  joined  by  oblique  lines  in  the 
way  shown.  For  beginners,  it  is  always  well  to  have  the 
points  set  off  both  on  the  top  and  bottom  lines,  but  it  is  really 
only  absolutely  necessary  to  have  them  on  the  top  line,  as 
the  experienced  hand  will  place  the  set-square  true  with  TE, 
and  having  drawn  that,  will  slip  it  along  in  the  way  we 
• have  already  explained  in  dwelling  on  the  uses  of  the  set- 
square,  and  draw  all  the  other  required  lines  parallel  to  the 
first  from  the  succeeding  points  on  the  top  line. 

193.  If  our  explanations  have  been  followed  up  to  this 
point,  our  readers  will  have  little  diflEiculty  in  the  application 
of  the  principle  of  the  diagonal  scale  to  practice.  Press  of 
space  has  compelled  us  to  only  show  one  distance,  i.e.,  from 
B to  C,  to  the  right  of  zero,  but  in  practice  we  should  mark 
off  some  four  or  five  spaces  each  equal  to  o,  i.  We  will  now 
assume  that  we  require  to  find  the  measurements  .7  inch,  1.4 
inch,  .38  inch,  and  1.63  inch  on  our  scale.  As  the  distance 
from  A to  B represents  an  inch  and  is  divided  into  ten  parts, 
if  we  count  seven  of  these  from  B,  we  obtain  the  first  measure- 
ment required ; it  is  B 7 in  our  figure.  The  next  distance, 
1.4  inch,  is  shown  by  the  distance  0 4 in  our  illustration.  The 
next  requirement,  .38,  is  the  thirty-eighth  of  an  inch  divided 
into  one  hundred  parts.  It  is  shown  by  FH,  three  complete 
divisions,  equal  to  three  tenths  or  thirty  hundredths,  being 
taken,  and  eight  tenths  of  another  part  being  added  to  them. 
As  the  number  we  have  required  ends  with  eight,  we  count  up 


CONSTRUCTION  OF  THE  DIAGONAL  SCALE,  91 


to  the  eighth  line  from  E,  and  then  measure  along  it.  Had  we 
wished  .37  instead,  we  should  have  stopped  one  below  point  H, 
and  reckoned  on  that  line  instead  the  three  complete  tenths 
and  the  seventh  of  the  remaining  tenth.  The  last  measure- 
ment, 1.63,  is  shown  by  JK.  To  obtain  it  we  count  up  three 
lines  from  E,  set  one  leg  of  the  compasses  at  K for  the  one 
inch  and  the  other  at  J,  six  complete  tenths  for  the  .60  and 
the  small  space,  equal  three  tenths  of  one  of  these  spaces,  for 
the  remainder. 

194.  A little  practice  should  render  the  use  of  the  diagonal 
scale  very  easy,  complex  as  it  may  appear  when  a descrip- 
tion merely  is  read.  Our  readers  should  not  rest  content, 
however,  with  any  mere  verbal  account,  but  endeavour  to 
make  scales  of  this  character  for  themselves,  taking  any 
three  consecutive  values,  as  miles,  furlongs,  and  chains, 
perches,  yards,  and  feet,  and  the  like,  and  then  setting  them- 
selves various  measurements  to  find  out.  Let  five  furlongs, 
for  example,  be  represented  by  an  inch,  and  then  find  a line 
2 miles  2 furlongs  3 chains  long;  or  a perch  being  repre- 
sented by  three  inches,  let  a line  3 perches  2 yards  i foot 
be  obtained. 

195.  Besides  the  line  of  chords,  many  other  mathematical 
lines  may  be  found  marked  on  the  scales  supplied,  but  most 
of  these  will  be  very  rarely,  if  ever,  used  by  the  ordinary 
draughtsman.  The  lines  of  sines,  tangents,  and  secants  are 
used  for  the  various  projections  of  the  lines  of  the  sphere, 
the  meridians,  parallels  of  latitude,  &c.  The  line  of  rhumbs 
is  employed  to  lay  down  or  track  a ship’s  course  on  the 
chart  and  to  calculate  the  run  made,  and  the  line  of  longi- 
tudes is  in  the  same  way  employed  in  the  science  of  navi- 
gation, and  a description  of  its  application  would  be  foreign 
to  our  present  aim. 


( 92  ) 


CHAPTER  IX. 

The  Marqnois  scale,  its  construction  and  examples  of  its  use — Cost — 
Natural  and  artificial  scale — Section  lines — Proportional  compasses 
— Scales  of  line — Enlargement  or  reduction  of  drawings — Scale 
of  circles — Division  of  circles  into  equal  parts — Scale  of  plans — 
The  determination  of  areas — Scale  of  solids— Determination  of 
bulks — Cost  of  proportionals — Wholes  and  halves — Eidograpli — 
Pantagraph — Measurement  of  angles — The  protractor — Division  of 
the  circle  into  degrees,  minutes,  and  seconds— Similar  and  equal 
figures — Line  of  chords. 

196.  We  have  yet  to  consider  the  Marqnois  scale,  the 
proportional  compass,  and  other  means  of  dividing  and 
measuring  lines,  and  to  the  first  of  these  we  now  turn  our 
attention. 

197.  The  Marqnois  scale,  so  called  from  the  name  of  its 
inventor,  is  rarely  used  except  for  military  drawing.  As, 
however,  the  name  figures  in  every  mathematical  instrument 
catalogue,  and  there  is  really  no  reason  why  the  things  should 
not  have  a more  extended  use,  we  give  some  little  space  to  a 
consideration  of  them. 

198.  A set  of  Marqnois  scales  consists  of  two  flat  rulers 
and  a triangle,  and  these,  from  their  bulk,  are  ordinarily 
supplied  in  a box  by  themselves,  the  price  for  such  a set 
being  about  eight  shillings.  Each  rule  is  a foot  long  and 
has  four  scales  on  each  face,  sixteen  scales  in  all,  of  which 
eight — the  inner  eight — are  called  natural  scales,  while  the 
others — those  on  the  outer  edges — are  termed  artificial  scales. 


USE  OF  THE  MARQUOIS  SCALE. 


93 


The  natural  scales  give  respectively  20,  35,  30,  35,  40,  45, 
50,  and  60  divisions  to  the  inch,  and  the  artificial  scales  that 
pair  off  with  these  on  each  edge  have  in  every  case  their 
divisions  three  times  as  large  as  those  of  the  natural  scale 
with  which  they  are  associated.  Each  artificial  scale  has  a 
zero  point  in  the  middle  of  it,  and  the  numbers  run  right 
and  left  from  this.  The  set-square  that  accompanies  these 
scales  is  a right-angled  triangle,  its  hypothenuse  being 
exactly  three  times  as  long  as  the  shortest  side ; it  bears, 
therefore,  the  same  proportion  to  it  as  the  artificial  scales  do 
to  the  natural.  A short  line  headed  by  a star  or  fleur-de-lys 
is  drawn  about  the  middle  of  the  hypothenuse,  and  the  longer 
of  the  two  remaining  sides  is  bevelled  down  for  convenience 
of  drawing  lines  either  in  pencil  or  ink.  All  the  instruments 
are  made  of  thick  boxwood,  and  will  stand  a good  deal 
of  usage.  The  principle  involved  in  their  construction 
is  based  on  the  second  proposition  of  the  sixth  book  of 
Euclid. 

199.  The  Marquois  scales  and  angle  together  will  often 
greatly  facilitate  work,  producing  various  constructions  with 
much  neatness,  accuracy,  and  rapidity.  They  are  to  some 
degree  at  once  the  equivalents  of  set-square,  straight-edge, 
parallel  ruler,  and  dividers,  and  the  student  will  readily  dis- 
cover many  ways  of  rendering  them  useful.  If,  for  example, 
we  desire  to  draw  two  parallel  lines  at  a distance  of  -||- 
apart,  we  draw  the  first  line  in  the  required  position  and 
place  the  edge  of  middle  length  of  the  set-square  in  contact 
with  it  throughout  its  length ; we  now  place  the  ruler  liaving 
the  scale  of  35  parts  to  the  inch  on  it  against  the  hypothenuse 
of  the  triangle,  so  that  its  zero  point  coincides  with  the 
index  line  on  the  set-square,  and  we  then  slide  the  set-square 
along  the  edge  of  the  scale  until  the  index  point  coincides 
with  the  thirteenth  division  from  zero.  The  line  we  are 


94 


MA  THEM  A TIC  A L INSTR  UMENTS. 


then  enabled  to  draw  will  be  parallel  to  the  first  and  at  the 
required  distance  from  it.  As  the  divisions  in  the  artificial 
scale  are  three  times  their  nominal  size,  the  measurements 
are  more  readily  and  accurately  taken,  and  any  slight  error  is 
proportionately  reduced,  so  that  what  appears  to  be  a hair’s- 
breadth  out  is  really  only  a third  of  that  distance  wrong. 
To  draw  a line  perpendicular  to  another,  we  place  the  shortest 
edge  of  the  set-square  to  the  first  line,  and  then  place  the 
ruler  against  the  longest  side.  We  then  slide  the  set-squaie 
along  the  ruler  until  its  third  edge  cuts  the  line,  and  any 
line  then  drawn  will  be  perpendicular  to  the  first.  For 
practice,  the  student  may  draw  two  lines  ^ J apart  at  right 
angles  to  two  others  apart. 

200.  We  have  in  fig.  47  taken  four  spaces;  the  first  of 
these  we  have  divided  into  five  equal  parts,  the  second  into 
seven,  the  third  into  fifteen,  and  the  last  into  twenty-five,  the 
wmrk  all  through  being  done  by  the  Marquois  scales  alone. 
For  the  first  of  these  we  employ  the  ''  40  scale ; we  draw 
the  first  line  with  the  bevelled  edge  of  the  set-square,  then 
place  the  scale  against  the  hypothenuse,  so  that  the  zero  on 
the  scale  and  the  index  point  on  the  triangle  coincide,  or 
read  into  each  other,’'  as  it  is  technically  termed.  We  then 
slip  the  set-square  along  the  edge  of  the  ruler,  and  as  the 
index  reads  into  every  eighth  point  along  the  artificial  scale 
from  zero,  we  draw  a line,  because  there  are  eight  fives  in 
forty.  If  we  had  wanted  twenty,  we  should  have  taken  every 
other  one;  if  eight,  every  fifth ; if  ten,  every  fourth.  In  the 
second  example  we  have  used  the  ''  35  scale,  and  drawn  a 
line  at  every  fifth  point.  To  get  the  third  series,  the  fifteen 
lines  in  the  given  space,  we  have  taken  the  “45,”  and  then 
drawn  a line  at  every  third  point  on  the  scale.  It  is  evident 
that  we  might  also  have  obtained  it  by  using  every  other  line 
of  the  “ 30,”  or  every  fourth  line  of  the  ''  60  ” scale.  In  the 


PROPORTIONAL  COMPASSES. 


95 


last  example,  twenty-five  divisions  in  the  given  space,  we 
have  used  the  ‘'50’’  scale,  and  taken  every  other  division  on 
its  edge.  Our  readers,  when  they  have  once  grasped  the 
princijjle  and  method  of  working,  will  find  no  difficulty  in 
multiplying  examples  indefinitely. 

201.  One  of  the  great  difficulties  of  the  novice,  as  we  have 
already  seen  in  our  remarks  on  the  'common  set-square  of 
45°,  is  the  even  drawing  of  a series  of  section  or  other  lines. 
Though  the  student  of  riper  experience  finds  little  difficulty 
in  drawing  them  by  the  unaided  eye,  the  beginner  often  gets 
them  very  irregularly  spaced  out.  It  will  be  obvious  on 
reflection  that  the  Marquois  scale,  or  any  other  evenly  divided 
ruler  that  has  its  divisions  coming  up  to  the  edge,  could  be 
employed  as  a guide,  the  set-square  being  slipped  along  it 
and  stopped  at  each  division.  All  the  lines  so  drawn  must 
necessarily  be  equidistant. 

202.  Proportional  compasses.  This  useful  and  ingenious 
instrument  is  composed  of  two  flat  and  similar  pieces  of 
metal,  each  having  a point  at  both  ends,  and  held  together 
by  a sliding  piece  that  travels  freely  up  and  down  the  central 
and  hollowed-out  portion  of  them,  or  by  means  of  a screw  can 
be  held  firmly  in  any  position  when  required.  Proportional 
compasses  are  used  to  enlarge  or  reduce  one  drawing  from 
another,  so  that  all  the  lines  of  the  example,  or  the  solids  and 
areas  expressed  by  them,  shall  all  bear  in  the  reproduction 
any  desired  proportion  or  ratio.  They  can  also  be  used  for 
dividing  circles  into  any  number  of  equal  parts  up  to  twenty, 
a problem  that  often  arises  when  polygons  have  to  be  drawn, 
or  the  positions  of  the  arms  or  teeth  of  wheels  marked,  and 
square  and  cube  roots  can  be  determined. 

203.  Our  readers,  on  taking  the  instrument  in  their  hands, 
will  notice  that  on  its  four  plain  surfaces,  the  long  strips  that 
on  each  side  of  the  instrument  bound  the  open  central  portion. 


96 


MATHEMATICAL  INSTRUMENTS. 


various  scales  are  engraved.  These  are  figured  and  named, 
those  on  one  face  of  the  compass  being  marked  as  lines  and 
circles,  while  the  other  two,  visible  on  turning  the  instru- 
ment over,  are  the  scales  for  plans  and  solids.  To  prepare 
the  instrument  for  use,  or  ''  set  ’’  it,  as  it  is  technically  termed, 
all  that  is  necessary  is  to  close  it  accurately,  so  that  each 
pair  of  legs  covers  each  other,  and  then  move  the  slider  until 
the  line  engraved  across  it  reads  into  the  required  division 
upon  any  of  the  scales.  Having  got  this,  all  that  remains  is 
to  tighten  the  screw,  and  the  instrument  is  ready  for  use. 

204.  The  most  ordinary  use  of  the  proportional  compass  is 
the  reduction  or  enlargement  of  drawings,  an  operation  that 
it  performs  with  beautiful  accuracy  and  celerity,  and  entirely 
doing  away  with  the  tedious  process  of  taking  every  measure- 
ment off  first  from  the  original,  and  then  from  the  enlarged 
or  reduced  scale.  The  modus  operandi  is  as  follows  : — The 
scale  marked  lines  is  taken,  and  the  line  of  the  slider  moved 
until  it  agrees  with  the  required  number,  as,  for  example,  4; 
the  instrument  is  then  tightened  up,  and  any  measurement 
taken  will  be  as  i is  to  4.  If,  then,  we  use  one  pair  of  points, 
as,  for  instance,  the  longer  legs  of  the  compass,  and  apply 
them  to  any  measurement  in  the  example,  the  other  pair  of 
points  will  always  mark  one-fourth  of  this.  On  the  other 
hand,  if  we  take  the  measurements  with  the  other  end  of  the 
compass,  the  second  pair  of  points  will  always  show  distances 
equal  to  four  times  those  taken.  The  ratio  between  the  mea- 
surements will  always  be  as  4 to  i,  and  we  can,  as  we  choose, 
use  either  end  of  the  instrument  to  take  the  first  measure- 
ments with,  and  the  alternative  end  will,  at  our  option,  give 
us  either  a distance  a fourth  of  this  or  four  times  as  great. 

205.  To  divide  the  circumference  of  a circle  into  any 
number  of  equal  parts  not  exceeding  twenty,  w^e  turn  to  the 
scale  of  circles  and  set  the  slider  to  the  required  number. 


THE  SCALE  OF  PLANS, 


97 


We  then  open  the  longer  legs  to  the  length  of  the  radius  of 
the  given  circle,  the  distance  apart  that  this  opens  the  shorter 
legs  being  the  required  length.  If,  for  example,  we  let  the 
slider  read  into  the  figure  8 on  the  scale,  and  set  the 
lono-er  le^s  to  the  radius  of  the  circle  that  we  wish  to  divide, 
the  opening  made  by  the  shorter  legs  of  the  compass  will  go 
just  eight  times  round  this  circle. 

206.  It  may  at  first  sight  appear  a disadvantage  that  the 
scale  of  circles  should  only  be  available  up  to  twenty  divi- 
sions, but  practically  this  drawback  need  not  be  much  felt. 
If,  for  instance,  we  want  to  divide  the  circumference  of  a 
circle  into  twenty-one  equal  parts,  it  would,  at  all  events, 
be  a clear  gain  of  time  to  divide  it  first  accurately  by  means 
of  the  proportional  compass  into  seven  equal  parts,  and  then 
by  trial  to  divide  one  of  these  sevenths  into  three  equal 
parts.  The  distance  once  found  would  readily  be  applied  to 
all  the  other  sevenths.  To  obtain  twenty-two  parts,  we  should 
first  find  eleven  and  then  bisect  each  part,  or  we  might  still 
more  expeditiously  arrive  at  our  result  by  drawing  two  radii 
at  right  angles  to  each  other,  and  taking  the  point  where 
each  touched  the  circumference  as  the  starting-points  of  two 
series  of  elevens.  To  obtain  thirty  divisions,  we  might  either 
obtain  fifteen  and  then  bisect  them,  or  else  find  the  distance 
that  would  be  correct  for  a pentagon,  and  set  this  off  round 
the  circle  six  times,  starting  successively  from  one  of  the  six 
points  that  we  could  first  so  readily,  by  means  of  the  radius, 
divide  the  circumference  into.  We  need  not  multiply 
examples,  for  we  are  sure  that  a little  reflection  on  the  part 
of  the  beginner  will  readily  indicate  what  course  to  pursue 
in  most  cases. 

207.  The  scale  of  plans  is  exceedingly  useful  where  it  is 
required  to  draw  areas  in  any  given  ratio.  Various  geometri- 
cal methods  are  available  for  this  purpose,  but  the  propor- 

G 


98  MA  THEM  A TICAL  INSTR  UMENTS, 


tional  compass  produces  the  result  at  once.  To  effect  this  we 
bringj  the  slide  down  until  the  line  on  it  reads  into  the 
required  line  on  the  scale.  If  we,  for  example,  set  it  at  figure 
5,  a square  or  triangle  having  a side,  or  a circle  having  a 
radius,  represented  by  the  width  we  open  the  larger  pair  of 
legs  will  be  five  times  the  area  of  a similar  figure  pricked  off 
by  the  shorter  pair  of  legs.  We  can  at  our  pleasure  use 
either  pair  of  points  for  our  first  measurement,  and  then  the 
second  pair  will  either  give  us  an  area  of  one  fifth  the  first 
figure  or  of  five  times  its  area. 

208.  The  remaining  scale,  that  of  solids,  is  employed  when 
we  wish  to  produce  drawings  that  shall  give  any  given  ratio 
between  the  capacities  of  two  bulks  or  solids  of  similar 
character.  As  the  way  of  setting  the  instrument  is  the  same 
throughout,  we  need  not  detail  it,  but,  as  an  illustration  of 
its  use,  we  may  suppose  that  we  have  made  a drawing  to 
measurement  of  a timber  stack,  and  that  we  wish  to  find  out 
how  large  another  of  five  times  the  cubical  contents  would 
be,  or  how  much  smaller  a pile  of  one-fifth  its  bulk  would 
appear.  Having  adjusted  our  slider  to  figure  5,  the  alter- 
native pair  of  points  to  those  applied  to  the  measurements 
in  the  drawing  would  at  once  give  the  necessary  increase 
or  decrease. 

209.  Our  readers  who  propose  to  provide  themselves  with 
an  instrument  so  useful  can  scarcely  complain  of  the  cost 
being  somewhat  high.  On  referring  to  a maker’s  catalogue, 
we  find  that  the  cheapest  six-inch  proportional  compass  costs 
a little  over  a sovereign,  and  a nine-inch  instrument  more 
than  as  much  again.  While  the  busy  man  of  large  practice 
and  scanty  leisure  will  at  once  avail  himself  of  so  valuable  an 
instrument,  for  its  services  to  him  will  soon  repay  its  cost,  the 
novice,  whose  needs  are  not  so  pressing,  nor  purse  possibly  so 
well  lined,  will  do  well  to  defer  the  purchase  awhile.  Every- 


METHODS  OF  ENLARGEMENT  OR  REDUCTION  99 


tiling  that  the  proportionals  can  do  can  be  done  by  other 
means,  and  it  will  do  the  beginner  no  harm  to  have  to  learn 
the  different  geometrical  methods  and  principles  by  which 
various  problems  have  to  be  met  by  those  who  do  not  pos- 
sess this  instrument. 

210.  Wholes  and  halves,  or  bisecting  compasses,  as  thej, 
are  sometimes  called,  are  sometimes  useful,  but  they  are 
costly  in  proportion  to  their  use,  and  the  student  would 
do  better  to  procure  a pair  of  proportionals  instead.  Wholes 
and  halves  are  like  a big  and  a little  pair  of  compass  points 
joined  together  by  a head  common  to  them  both.  This  head 
is  one-third  of  the  distance  from  one  pair  of  points  to  the 
other,  so  that  whatever  distance  we  open  the  little  pair,  the 
other  pair  are  opened  as  much  again,  and  whatever  distance, 
conversely,  we  open  the  larger  pair,  the  small  pair  open  to 
half  as  much. 

21 1.  Where  it  is  required  to  make  plans  either  half  or 
twice  the  size  of  an  original  drawing,  these  compasses  would 
at  once  be  available,  but  practically  so  many  other  propor- 
tions are  required  for  which  wholes  and  halves  are  of  no 
service,  that  we  can  only  reiterate  our  advice,  and  urge  the 
beginner  to  get  proportionals  instead.  A very  slight  in- 
crease of  cost  will  give  an  instrument  that  will  do  all  that 
the  wholes  and  halves  will,  plus  a great  deal  more.  The 
chief  value  of  wholes  and  halves  is,  as  their  second  name 
implies,  for  making  bisections.  This  they  undoubtedly  do 
very  quickly  and  accurately,  but  whether  in  most  cases  it  is 
worth  while  to  spend  a sovereign  in  an  instrument  to  do  this 
opens  up  another  question  entirely. 

212.  The  eidograph  and  pantograph  are  other  ingenious 
contrivances  for  enlarging  or  reducing  drawings,  but  both  lie 
beyond  the  sphere  of  the  beginner’s  operations.  Should 
he,  however,  desire  to  possess  either  or  both  of  these,  the 


lOO 


MATHEMATICAL  INSTRUMENTS. 


prices  in  each  case  range  from  about  twelve  to  sixteen 
pounds. 

213.  Two  or  three  instruments  for  obtaining  angles  now 
remain  to  be  mentioned.  The  one  in  most  ordinary  use  is 
the  protractor.  We  have  in  fig.  33  a representation  of  the 
commonest  form,  while  another  is  shown  in  fig.  48. 

214.  Circles,  irrespective  of  their  size,  are  in  all  kinds  of 
mathematical  work  considered  to  be  divisible  into  360  equal 
parts ; each  of  these  parts,  or  degrees,  as  they  are  technically 
termed,  is  divided  again  into  60.  This  secondary  division  is 
in  turn  divided  in  like  manner ; so  that,  put  into  table-book 
form,  the  facts  would  run  as  follows  : — 60  seconds  one  minute, 
60  minutes  one  degree,  and  360  degrees  in  a circle.  In 
mathematical  drawing  we  rarely,  however,  get  beyond  degrees 
and  half-degrees,  and  the  division  into  seconds  is  for  this  pur- 
pose purely  theoretical. 

215.  As  it  would  be  very  tiresome  to  be  always  obliged  to 
write  the  word  degrees  ” after  the  figures  marking  them,  a 
sign  has  been  adopted  instead.  A small  circle  is  placed 
after  the  number,  and  level  with  the  top  of  it,  so  that  72 
degrees,  for  example,  would  be  always  written  72°. 

216.  Both  the  rectangular  protractor,  fig.  33,  and  the  semi- 
circular protractor,  fig.  48,  are  figured  for  180°,  and  this  is 
ordinarily  all  that  is  required.  Where  more  than  this  is 
wanted,  a circular  protractor  is  required ; on  this,  of  course, 
all  the  360°  are  marked.  If  our  readers  will  notice  our  two 
illustrations,  they  will  see  that  the  bottom  edge  is  quite 
plain  except  that  in  its  centre  it  has  one  line.  This  line  is 
ordinarily  marked  by  a star  or  some  such  device,  to  render  it 
more  conspicuous.  Around  all  the  other  edges  of  the  rect- 
angle or  the  arc  of  the  second  form  of  protractor  a series  of 
radiating  lines  will  be  perceived.  These  lines,  if  continued, 
would  all  meet  at  the  outer  end  of  the  star-marked  line. 


THE  MEASUREMENT  OF  ANGLES. 


lOI 


Every  tenth  line  is  numbered,  and  the  fifth  or  half-way  line 
between  each  of  these  figured  lines  is  made  a little  longer — a 
sufficient  guide  to  the  eye. 

217.  When  any  angle  has  to  be  set  off  from  a given  point, 
the  protractor  is  placed  so  that  its  lower  edge  coincides  with 
the  first  line  of  the  required  angle,  and  having  the  central 
point  on  this  line  accurately  placed  at  the  spot  where  the 
angle  has  to  be  made.  The  eye  then  runs  along  the  line  of 
figures  until  the  desired  angle  is  found,  when  a pencil-tick 
is  made  at  this  point,  and  a line  joining  it  with  the  desired 
starting-point  of  the  angle  is  drawn.  This  second  line  makes 
wdth  the  first  the  required  slant  or  angle. 

218.  Beginners  are  often  puzzled  because  the  same  line  on 
the  protractor  bears  two  very  different  numbers ; thus  one  is 
marked  as  30°  and  150°,  or  10°  and  170°,  or  again  80°  and  100°. 
A glance  at  fig.  49  will,  however,  explain  this.  For  it  will 
be  seen  that  while  at  point  A we  draw  the  required  angle, 
BAG,  of  30°,  we  at  the  same  time  create  another  angle,  DAB, 
and  this  angle  is  150°.  The  total  of  the  two  angles  will 
always  make  180°  A very  little  practice  will,  however, 
enable  a beginner  to  see  which  is  the  line  he  really  wants. 

219.  The  three  angles  of  any  triangle  when  added  together 
should  make  180°.  In  the  set-squares  we  see  that  one  of 
them  has  one  angle  of  90°  and  the  other  two  of  45°,  while 
the  other  common  form  has  them  of  30°,  60°,  and  90°.  In 
each  case  the  total  is  180°.  We  can,  therefore,  construct  an 
equilateral  triangle  by  means  of  the  protractor.  For  if  it  is 
equal-sided,  it  must  also  be  equal-angled,  and  these  angles 
must  then  each  of  them  be  one  third  of  180°,  ie.,  60°.  The 
beginner  will  find  it  good  practice  to  draw  some  irregular 
triangles,  and  then  measure  their  angles  by  means  of  the 
protractor:  as  he  knows  what  the  total  should  be,  he  can 
readily  add  his  results  together,  and  see  how  far  his  readings 


102 


MATHEMATICAL  INSTRUMENTS. 


are  correct.  The  product  of  90,  40,  and  50  would  show  a 
correct  reading ; the  product  of  80,  90,  and  1 2 an  error.  If, 
then,  we  know  by  measurement  what  any  two  angles  of  a 
triangle  are,  it  is  superfluous  to  measure  the  third. 

220.  We  will,  suppose  that  we  desire  to  know  what  angle 
two  lines  would  meet  at  if  continued,  the  actual  continuation 
being  for  some  reason  impracticable.  The  problem  is  pre- 
sented to  us  in  fig.  50.  EF  and  HI  are  two  lines  that  converge, 
but  the  point  of  convergence  falls  beyond  the  paper.  All 
that  is  necessary  is  to  draw  a third  line,  JK,  across  them 
anywhere  in  their  length  ; find  what  the  sum  of  the  two  inner 
angles  would  be,  and  the  difference  between  that  and  180"^ 
would  give  the  angle  at  which  lines  EF  and  HI  would  meet. 
In  referring  to  any  angle,  its  apex  is  denoted  by  the  central 
letter;  thus  in  fig.  50  we  say  angle  KJF, because  J is  the 
point  at  which  the  lines  KJ  and  EJ  meet;  it  would  be  alto- 
gether misleading  to  call  it  JKF,  JFK,  KFJ,  or  FKJ.  Bear- 
ing this  in  mind,  we  are  now  able  to  refer  to  the  angles  in  our 
figure.  On  testing  these  with  the  protractor,  we  find  that 
IKJ  is  1 1 3""  and  KJF  35j°;  the  lines  EF,  HI,  must  then  be 
inclined  to  eaeh  other  at  an  angle  of  3iJ° 

221.  Eectangular  protractors  are  generally  made  six  inches 
long,  and  either  of  wood  or  ivory,  while  the  semicircular  or 
circular  ones  are  made  ordinarily  of  brass,  though  electrum  is 
sometimes  employed.  The  price  of  an  ordinary  six-inch  box- 
wood protractor  should  be  about  eighteenpence,  and  of  a brass 
one  about  twice  as  much.  Horn  and  paper  are  sometimes 
employed.  A six-inch  semicircular  horn  protractor  would 
cost  about  a shilling.  Their  transparency  is  sometimes  an 
advantage,  as  they  do  not  obscure  the  drawing  they  are 
placed  over,  and  one  risk  or  error  in  joining  wrong  points  is 
thus  avoided,  but  they  are  rather  given  to  cockling  up.  The 
brass  semicirculars,  as  may  be  seen  in  our  figure,  have  a large 


Vl!l 


m UB^ARlf 
OF  THE 


THE  USE  OF  THE  PROTRACTOR. 


103 


opening  in  the  centre,  and  this  is  practically  as  safe  as  the 
transparency  of  the  horn. 

222.  Paper  protractors  are  somewhat  different  in  character. 
They  are  printed  on  square  cardboard,  and  give  an  entire 
circle.  Instead  of  marking  off  the  degrees  from  the  outer 
edge,  as  in  the  other  kinds,  they  are  marked  off  from  the 
inner,  the  whole  of  the  interior  being  cut  away.  The  general 
look  of  the  thing,  therefore,  is  that  of  a card  about  a foot 
square,  and  having  a large  round  hole  in  the  middle  of  it ; 
all  round  the  edge  of  this  opening  the  degrees  are  marked. 
As  the  central  point  from  which  all  the  lines  radiate  is  cut 
away,  another  method  of  reading  off  the  degrees  has  to  be 
employed.  Two  lines  are  drawn  at  right  angles  to  each  other, 
by  means  of  the  T square  and  set- square,  through  the  point 
at  which  the  angle  has  to  be  drawn,  and  these  lines  are  made 
long  enough  to  reach  the  divided  edge  of  the  protractor.  The 
insumment  is  then  shifted  about  until  one  line  coincides  with 
the  0°  and  the  1 80°,  and  the  other  into  the  two  90°.  The 
onl)^  advantage  that  this  form  possesses  is  that  a series  of 
lines  at  various  angles  can  be  drawn  from  this  central  point 
without  having  to  move  the  protractor  away  each  time  a 
line  is  drawn. 

223.  Any  triangle,  we  have  seen,  will  have  the  sum  of  its 
angles  1 80° : if  our  readers  care  to  increase  their  practice  in 
the  measurement  of  angles,  they  may  next  proceed  to  measure 
quadrilateral  or  four-sided  figures.  All  four-sided  figures 
must  also  have  four  angles,  and  the  sum  of  these  angles  will 
always  be  equal  to  360°.  Squares  and  oblongs  are  rectangu- 
lar figures ; all  their  angles  are  alike,  and  therefore  each  is  90°, 
the  quarter  of  360°.  The  other  quadrangular  figures  are  the 
rhombus,  rhomboid,  trapezium,  and  trapezoid. 

224.  All  the  angles  of  a regular  polygon  are,  like  its  sides, 
equal,  and  their  value  is  constant.  Any  nonagon,  for  ex- 


104 


MATHEMATICAL  INSTRUMENTS. 


ample,  will  always  have  its  adjacent  sides  making  the  same 
angle  with  each  as  any  other  nonagon,  irrespective  of  size, 
and  this  angle  will  never  he  found  in  an  octagon  or  decagon. 
Every  polygon  has  its  special  angle,  so  that  if  only  two  sides 
and  one  angle  are  given  we  could  construct  the  whole  figure. 
Practically,  however,  polygons  are  never  drawn  by  the  mea- 
surement of  their  angles,  so  we  need  not  give  in  detail  the 
specific  angle  of  each  kind. 

225.  The  measurement  of  angles  enters  very  largely  into 
trigonometry  and  land  surveying.  By  means  of  these  measure- 
ments we  are  able  to  reproduce  any  irregular  right  line  form, 
either  the  size  of  the  original  or  with  any  degree  of  enlarge- 
ment or  diminution.  An  example  of  this  is  given  in  fig.  51. 
ABCDEEG-  is  an  irregular  plot  of  ground,  and  HUKEMN" 
the  same  plot  to  a larger  scale.  To  effect  this,  the  origiaal 
figure  must  be  cut  up  into  any  convenient  arrangement  of 
triangles,  not  necessarily  those  shown  in  our  figure,  and  these 
are  reproduced,  triangle  by  triangle,  starting  from  the  rew 
base  line.  In  our  figure,  HI  is  the  representative  of  AB,  and 
whatever  angles  are  observable  at  the  extremities  of  AB  are 
also  drawn  at  the  extremities  of  HI ; angle  LHI  is,  therefore, 
similar  to  EAB,  and  angle  LIH  is  equivalent  to  ABE.  On 
line  HL  we  proceed  to  construct  a triangle,  HLhT  having 
angles  similar  to  those  of  the  triangle  AEG-,  and  on  line  LI 
we  draw  a triangle,  LKI,  having  its  angles  agreeing  with  tliose 
of  the  triangle  EBD.  We  proceed  in  this  matter  step  by 
step  until  the  whole  figure  is  constructed. 

226.  Beginners  are  sometimes  puzzled  by  the  difference 
drawn  between  the  words  ''similar”  and  "equal,”  where,  as  in 
the  case  of  two  triangles,  the  angles  are  in  the  two  figures  the 
same,  these  figures  are  said  to  be  similar.  Thus  EEG  and  LMH 
are  similar  triangles,  though  one  is  evidently  much  larger 
than  the  other;  were  they  identical  in  size  as  well,  they 


USE  OF  SCALE  OF  CHORDS. 


105 


would  be  equal  figures.  A square  on  a base  of  one  inch  is 
a similar  figure  to  a square  on  a base  of  six  inches,  but  we 
must  find  another  square  also  on  a base  of  one  inch  before 
w^e  can  say  that  we  have  a figure  equal  to  the  first.  In  the 
same  way  two  circles  6ach  struck  with  a radius  of  three 
inches  are  equal  circles,  while  two  circles  struck  with  radii 
of  two  inches  and  ten  respectively  are  similar.  HIJKLMisr 
is  a similar  figure  to  ABCDEFG. 

227.  On  many  scales  a line  of  divisions  marked  C”  or 
Cho  ” may  be  found ; it  may  be  seen,  for  example,  in  figs. 
33  and  34.  This  is  called  the  line  of  chords ; it  is  another 
method  of  obtaining  angles.  To  employ  it,  a curve  is  struck 
having  a radius  equal  to  the  distance  0-60  on  the  scale,  and 
on  this  arc  the  required  number  of  degrees  is  marked  by 
taking  the  distance,  whatever  it' may  be,  along  the  scale 
from  o and  then  transferring  this  distance  to  the  curve. 
When  the  angle,  as  is  ordinarily  the  case,  has  to  be  drawn 
from  a given  point  in  a given  straight  line  this  point  is  taken 
as  the  centre  from  which  the  arc  is  struck,  and  where  the 
arc  touches  the  straight  line  becomes  the  starting-point  for 
measuring  the  distance  upon  the  curve. 


( io6  ) 


CHAPTER  X. 

The  sector — Principle  of  its  construction — Line  of  lines — Illustrative 
examples  of  its  use — Enlargement  or  reduction  in  any  given  pro- 
portion— Line  of  polygons — Illustrations  of  its  use — Description 
of  polygons  or  multilateral  figures — The  line  of  chords — Examples 
of  its  use — Cost  of  the  sector — A box  of  mathematical  drawing 

o 

instruments — Second-hand  things  — School  sets — Cost  of  various 
selections  of  instruments — Long  measure — Surveyor’s  measure — 
Square  measure  — Solid  measure  — Abbreviations — Figuring  di- 
mensions on  drawings. 

228.  The  sector.  This  is  a very  ingenious  instrument,  and 
our  work  would  certainly  be  incomplete  if  we  failed  to  refer 
to  it,  for  it  is  frequently  placed  in  the  box  of  instruments 
supplied ; but  at  the  same  time  it  is  rarely  used,  as  other 
means  are  generally  available  for  obtaining  the  various 
results.  Sines,  scants,  tangents,  &c.,  are  readily  obtainable 
by  its  means,  but  as  these  are  rarely  required  in  draw- 
ing work,  the  instrument  is  put  aside  or  only  used  as  a 
ruler. 

229.  The  sector  consists  of  two  equal  rulers  jointed 
together  at  one  end.  These  rulers  or  legs  lie  side  by  side 
when  the  instrument  is  shut  up,  but  they  can  be  opened  to 
any  angle  with  each  other,  or  extended  so  far  that  they  make 
a straight  line.  A sector,  then,  of  six  inches  will,  when  fully 
open,  be  a foot  long,  and  the  outer  edges  are  on  one  side  of 
the  instrument  marked  with  inches,  so  as  to  enable  it  to  be 


THE  LINE  OF  LINES, 


107 


used  as  a foot-rule.  Various  scales  are  marked  upon  it,  but 
the  only  ones  we  need  here  refer  to  are  those  marked  L/' 
‘‘POL/’  and  ” Pig.  52  is  a representation  of  one  side 
of  a sector ; it  shows  two  of  the  scales  we  have  mentioned, 
and  others  would  be  found  on  the  other  side. 

230.  The  scale  marked  ‘"L”  is  the  line  of  lines.  It  is 
used  for  dividing  lines  into  equal  parts,  obtaining  propor- 
tionals, constructing  scales,  &c.  The  line  of  lines  is  on  each 
half  or  leg  divided  into  ten  primary  divisions,  and  each  of 
these  is  again  divided  into  ten  parts.  All  measurements 
taken  lengthways  on  any  of  the  scales  are  called  lateral  dis- 
tances, and  all  distances  taken  across  from  leg  to  leg  are 
termed  transverse.  If,  now,  we  want  to  divide  a line  three 
and  a half  inches  long  into  nine  equal  parts,  we  first  open 
the  sector  until  the  distance  from  9 to  9 is  equal  to  the 
length  of  the  line.  Then  from  one  extremity  of  the  line  we 
set  off  the  distances  8-8,  7-7,  6-6,  and  so  on,  and  the  line 
will  be  divided  as  desired.  The  proof  of  the  correctness  of 
this  will  be  found  in  the  second  problem  of  the  sixth  book 
of  Euclid. 

231.  It  is  required,  again,  to  find  .84  of  a line  three 
inches  long.  To  do  this  we  extend  the  legs  of  tlie  sector 
until  the  compasses  with  a radius  of  three  inches  have  their 
points  resting  on  the  two  lo’s.  These  primary  values,  we 
must  mention,  may  be  taken  either  as  units,  tens,  hundreds, 
thousands,  &c.,  as  we  choose,  and  as  the  problem  before  us  is 
to  find  the  eighty-fourth  part  out  of  a hundred,  we  consider  the 
entire  line  as  one  hundred  parts  long.  If,  now,  while  keep- 
ing the  sector  open  at  the  same  angle,  we  shift  the  compass 
until  its  points  rest  on  the  fourth  secondary  division  on  from 
the  eighth  primary,  we  have  found  a distance  which  is  .84  of 
the  whole  line. 

232.  The  fractions  need  not  be  decimals  ; we  can  as  readily 


io8 


MA  THEM  A TICAL  INSTRUMENTS. 


take  any  others.  We  will  suppose  that  w^e  want  to  find 
of  a line,  the  whole  line  being  two  inches  long.  To  work 
this  out  we  take  a distance  of  two  inches  by  compass  and 
extend  the  limbs  of  the  sector  until  the  points  of  the  com- 
pass rest  on  the  95  on  each  side;  we  then  close  the  compasses 
until  they  span  the  distance  between  the  two  7 1 points,  and 
this  distance  we  set  off  from  one  end  of  our  line ; it  is  the 
of  it.  We  can  in  like  manner  take  fractions  of  fractions.  The 
first  line  in  fig.  53  is  divided  at  B in  .43  of  its  length;  the 
second  line,  DE,  has  at  E a point  from  D ; while  the  third 
or  lowest  line,  GH,  is  divided  at  point  K into  | of  -f-  of  its 
length.  This  redivision  can  be  carried  to  any  extent.  The 
line  GH  is  first  taken  and  the  sector  opened  until  the  length 
of  the  line  spans  from  7 to  7.  The  distance  5-5  is  then  taken 
and  marked  off  from  G to  GI.  This  point,  I,  would  not  really 
be  shown,  but  we  have  marked  it  to  facilitate  our  description. 
What  is  really  wanted  is  | of  GI.  The  sector  is  now  taken 
and  extended  until  the  distance  GI  reaches  from  3 to  3 on 
the  scale,  and  then  the  distance  2 to  2 gives  us  the  required 
point  K.  GK  is  the  | of  of  GH. 

233.  A given  line,  LM,  on  a map  represents  41  miles,  and 
we  desire  to  make  a scale  that  will  be  available  up  to  100 
miles.  To  efiect  this,  we  first  take  the  given  distance,  and 
open  the  sector  until  the  extremities  of  our  line  agree  with 
the  first  division  beyond  the  4 on  each  side,  and  then  extend 
them  from  10  to  10.  This  latter  distance  is  the  100  miles 
required,  and  we  divide  this  anew  into  tens,  and  the  last 
tenth  to  the  left  into  single  miles,  so  as  to  make  the  whole 
into  a practicable  scale  for  measuring  any  distances  in  our 
map. 

234.  Drawings  can,  by  means  of  the  sector,  be  reduced  or 
enlarged  in  any  given  proportion.  If,  for  example,  we  desire 
to  reduce  a plan  in  the  proportion  of  5 to  9,  we  take  any  line 


THE  LINE  OF  LINES, 


109 


in  the  original  drawing,  and  open  the  instrument  until  the 
ends  of  the  line  rest  on  9 and  9.  We  now  take  the  distance 
between  5 and  5,  and  this  at  once  gives  us  the  length  of  the 
line,  the  equivalent  of  the  first,  in  our  reduced  drawing.  We 
could  proceed  in  like  manner  with  all  the  lines,  first  measur- 
ing them  from  9 to  9 and  then  from  5 to  5,  but  practically 
we  should,  having  once  got  the  proportion  between  the 
original  and  the  reproduction,  make  the  two  scales,  and  work 
from  one  to  the  other.  These  scales  would,  of  course,  be 
made  by  means  of  the  sector. 

235.  To  find  a third  proportional  to  two  given  lines.  Let 
these  lines,  say,  be  as  4 is  to  6.  We  take  then  laterally  along 
one  leg  of  the  sector  the  distance  from  o to  6,  and  open  the 
instrument  until  this  distance  agrees  with  4 and  4.  We  now 
take  the  transverse  distance  between  6 and  6 and  measure  it 
laterally  from  zero,  and  we  find  that  it  falls  upon  point  9,  a 
sufficient  proof  that  our  work  is  correct,  for  a moment’s  con- 
sideration will  show  us  that  as  4 is  to  6 so  is  6 to  9. 

236.  Many  other  examples  of  the  great  use  of  the  line  of 
lines  might  be  given,  but  enough  has  been  said  to  indicate 
the  mode  of  procedure,  and  our  readers  may  readily  find 
other  applications  of  the  principle.  The  true  line  of  lines  is 
the  inner  one  in  each  case.  It  will  be  seen  that  at  each 
point  in  our  illustration,  fig.  52,  there  are  three  parallel 
lines  close  together;  the  upper  two  are  merely  to  enclose 
spaces  in  which  the  divisions  of  the  scale  are  put,  the  tenths 
going  a little  beyond  the  outer  line,  the  fifths  just  reaching 
it,  and  the  intermediate  points  only  extending  from  the  inner 
to  the  middle  line.  The  true  line  of  lines  is  the  inner  one, 
as  we  have  said;  it  is  the  only  one  that  really  goes  to  the 
centre.  It  is  marked  at  its  outer  extremity  by  a small  brass 
nail-head,  and  is  in  many  instruments  ruled  in  red,  wliile  the 
other  two  parallel  to  it  are  in  black.  In  the  same  way  it  is 


no 


MATHEMATICAL  INSTRUMENTS, 


the  inner  line  again  that  is  the  true  line  of  polygons,  and 
this,  too,  is  marked  by  a brass  stud  at  its  outer  extremity, 
and  by  being  ruled  in  red. 

237.  We  proceed  now  to  see  what  use  may  be  made  of  the 
line  marked  ''  POL.’’  This  line  may  be  seen  in  our  illus- 
tration, fig.  52,  nearer  the  line  of  junction  of  the  two  legs 
than  the  line  we  have  hitherto  been  considering.  It  will  be 
remembered  that  the  radius  of  a circle  goes  six  times  rounds 
its  circumference,  and  the  scale  is  so  graduated  that  if  we 
take  any  given  distance  as  radius,  and  adjust  it  transversely 
from  points  6 and  6,  then  the  distance  5-5  would  go  just  five 
times  round  that  circle,  9-9  just  nine  times,  and  so  on.  The 
scale  is  marked  from  four  to  twelve,  and  in  fig.  54  we  have 
drawn  a circle,  and  placed  within  it  figures  of  four,  five,  six, 
seven,  and  eight  equal  sides,  doing  all  dividing  up  of  the 
circumference  by  means  of  the  sector  alone. 

238.  Though  we  find  the  first  figure  on  the  line  of  polygons 
to  be  a four,  the  smallest  number  of  sides  that  constitutes  a 
polygon  (or  many-angled  figure,  as  the  word  means),  is  ordi- 
narily in  geometrical  work  taken  as  five.  In  some  old  works 
the  triangle  is  called  the  trigon,  or  three-angled  figure,  and 
the  square  a tetragon,  or  four-angled  form ; but  neither  one  or 
other  of  these  could  legitimately  be  called  a polygonal  or 
many-angled  figure.  A polygon  may  have  any  number  of 
sides  beyond  four,  and  each  polygon  bears  in  addition  a 
specific  name,  indicating  what  the  number  of  sides  is.  Thus 
we  have  the  pentagon,  hexagon,  heptagon,  octagon,  nonagon, 
or  decagon,  according  to  whether  the  figure  has  five,  six, 
seven,  eight,  nine,  or  ten  sides. 

239.  We  do  not  ordinarily  go  much  beyond  these  numbers 
in  practice,  but  any  one  acquainted  with  the  Greek  names  of 
the  numerals  will  have  no  difficulty  in  naming  any  special 
form.  The  sixteen  or  six-and-ten-sided  figure  would  be  a 


IX. 


THE  imm 

OF  THE 

IISIVESSITV  CF  JLL!J:3!S 


POLYGONS  DRAWN  BY  AID  OF  SECTOR. 


Ill 


hexadecagon,  and  the  same  way  of  compounding  the  name 
would  be  applied  in  all  cases.  Sometimes,  where  one  may 
not  feel  quite  sure  of  the  real  name,  or  would  be  afraid  of 
being  suspected  of  affectation  in  using  it,  we  find  the  figure 
spoken  of  as  a polygon  of  so  many  sides,  sixteen,  eighteen,  or 
whatever  it  may  be. 

240.  Polygons  are  also  termed  multilateral  figures  ; the 
first  word,  we  have  seen,  means  many-angled,  and  the  second 
signifies  many-sided.  Either  are  equally  descriptive  and 
appropriate,  though  general  usage  has  made  the  former  the 
more  familiar  term. 

241.  When  a polygon  is  spoken  of,  it  is  always  understood 
to  mean  a regular  polygon,  i,e,,  one  having  all  its  sides  and 
all  its  angles  equal.  When  this  is  not  intended,  the  figure 
referred  to  is  distinctly  called  an  irregular  polygon ; such 
figures  may  have  their  angles  equal  and  their  sides  unequal, 
or  their  sides  may  be  equal  but  the  angles  unequal,  or  both 
may  be  unequal. 

242.  Having  described  the  simple  method  by  which,  by 
means  of  the  sector,  we  are  enabled  to  draw  any  polygon  up 
to  a dodecagon  in  any  given  circle,  we  now  proceed  to  con- 
sider how  the  converse  of  this — one  side  of  the  polygon  being 
given  to  construct  it — is  to  be  effected.  We  will  suppose 
that  we  wish  to  construct  an  octagon  having  sides  an  inch 
long.  We  begin  by  opening  the  sector  until  the  points  of 
the  compass,  open  to  the  required  length  of  side,  rest  on  the 
corresponding  opposite  numbers  in  this  case  on  the  8 and  8,  as 
our  figure  is  to  be  an  octagon.  We  then  take  the  distance 
that  we  find  points  6 and  6 apart,  and,  with  this  distance  as 
radius,  we  draw  arcs  from  each  extremity  of  the  given  side 
of  the  polygon  as  centres.  The  point  at  which  these  arcs 
intersect  is  the  centre  of  a circle  that  will  contain  the  given 
line  the  required  number  of  times,  the  radius  of  this  circle 


II2 


MATHEMATICAL  INSTRUMENTS. 


being  the  distance  from  the  intersection  of  the  arcs  to  either 
end  of  the  given  line. 

243.  We  now  pass  to  a consideration  of  the  line  marked 
''  C on  the  sector,  the  line  of  chords.  This,  like  the  line  of 
chords  on  the  protractor,  is  nsed  to  determine  any  given 
angle  by,  but  it  differs  somewhat  in  construction.  We  will 
suppose,  for  the  sake  of  illustration,  that  we  wish  to  make  an 
angle  of  43°.  Having  drawn  a line,  and  indicated  on  it  the 
point  that  is  to  be  the  apex  of  the  required  angle,  and  the 
starting-point  of  the  new  line  that  is  to  make  at  this  point 
the  required  angle  with  the  first  line,  we  draw  an  arc  of  any 
radius  from  this  point  as  centre,  and  having  one  extremity 
resting  on  the  given  line.  We  now  open  the  sector  until  the 
two  points  of  the  compass  rest  on  points  60,  60,  and  then, 
whatever  the  transverse  measurement  between  43  and  43 
may  be,  we  take  this  distance  and  set  it  along  the  arc.  The 
measurement  is  commenced  from  the  point  where  the  arc 
touches  the  first  straight  line,  and  wherever  the  radius  cuts 
the  arc  we  draw  a line  from  that  point  to  the  centre  from 
which  the  arc  was  struck.  The  two  straight  lines  will  be  at 
the  required  angle  with  each  other.  The  work  done  by  the 
sector  may  be  tested  and  proved  by  the  protractor. 

244.  As  half  degrees  are  marked  on  the  line  of  chords  on 
the  sector,  a little  care  must  be  exercised,  or,  when  we  think 
we  are  taking  43,  it  may  really  be  only  41^. 

245.  Sixty  is  the  highest  figure  given  in  the  sectoral  line 
of  chords.  When  we  wish  to  get  an  angle  of  more  than  60°, 
we  have  to  divide  the  number  by  two  or  three,  and  having 
obtained  the  half  or  the  third  of  it,  step  the  distance  twice  or 
thrice  along  the  arc. 

246.  The  line  of  chords  can  be  used  to  construct  any  poly- 
gon of  which  the  sides  will  divide  into  360  without  remain- 
der. Eight,  for  instance,  is  contained  just  forty- five  times  in 


USE  OF  THE  SECTOR. 


113 

this  number,  and  eighteen  just  twenty  times ; either  of  these 
figures,  therefore,  could  be  constructed  by  means  of  the  line 
of  chords,  and  a little  reflection  on  the  part  of  our  readers 
will  enable  them  to  add  others  to  these.  As  an  example  of 
the  modus  operandi,  we  will  consider  how  an  octadecagon 
AYOuld  be  constructed.  On  the  circle  being  struck  of  any 
required  size,  the  radius  is  taken,  and  the  legs  of  the  sector 
shifted  until  the  compass  points  rest  on  the  two  60  points  on 
either  side.  As  eighteen  is  contained  twenty  times  in  360, 
we  now  place  the  compass  points  on  20-20,  and  this  distance 
is  one  side  of  the  required  polygon.  All  that  now  remains  is 
to  mark  it  off  carefully  eighteen  times  round  the  circumfer- 
ence of  the  circle. 

247.  The  other  mathematical  lines  that  are  sometimes  put 
on  the  sector  we  need  not  here  enter  into,  since  they  are  for 
the  working  out  of  problems  that  drawing  does  not  ordinarily 
have  much  to  do  with.  We  trust  that  we  have  nevertheless 
brought  forward  enough  to  make  the  beginner  regard  his  sector 
with  more  respect  than  he  once  felt  for  it ; to  consider  it  at 
least  as  something  more  than  a jointed  foot-rule.  The  price 
of  a sector,  we  may  just  add  in  conclusion,  should  be  about 
eighteenpence  if  of  boxwood,  or  five  shillings  if  made  of 
ivory.  The  latter  are  much  to  be  preferred,  as  the  readings 
are  much  clearer.  Ivory,  too,  is  much  harder  than  box- 
wood, and  where,  as  in  the  case  of  the  60,  60,  and  other 
points  of  the  sector,  we  are  often  using  the  same  part  of 
the  instrument,  the  less  dense  character  of  the  wood  is  a 
disadvantage. 

248.  Having  now  gone  through  the  various  instruments  in 
detail  that  are  found  in  ordinary  use,  we  may  devote  a few 
lines  to  the  consideration  of  what  they  form  in  the  aggregate ; 
what  we  understand  when  we  hear  a box  of  mathematical 
instruments  referred  to. 


H 


MATHEMATICAL  INSTRUMENTS. 


114 


249.  Two  or  three  very  inferior  instruments  may  often  be 
seen  fastened  together  on  a piece  of  cardboard  and  retailed  to 
the  unwary  for  a shilling  or  so.  Such  things  cannot  possibly 
be  good,  and  any  one  who  really  intends  to  do  any  real  work 
should  carefully  eschew  them. 

250.  Second-hand  instruments  may  at  times  be  procured, 
but  too  often  their  purchase  proves  no  economy ; the  joints 
are  out  of  order,  screws  missing,  scales  split,  or  divers  other 
indications  that  the  things  are  worn  out.  It  is  a well-known 
fact,  too,  that  large  numbers  of  inferior  things  are  made  by 
unscrupulous  manufacturers,  and  then  got  into  the  market 
as  second-hand  goods.  We  recall  the  experience  of  a friend 
of  ours  who  once  got  a note  from  a pawnbroker,  saying  that 
he  had  several  sets  of  instruments  by  him,  and  that  he  should 
be  glad  to  dispose  of  them.  They  were  seen  and  purchased, 
a wonderful  bargain ; but  when  they  were  given  out  to  his 
pupils,  he  realised  too  late  what  an  unsatisfactory  lot  of 
things  had  been  palmed  off  on  him. 

251.  Where  economy  is  a consideration,  it  is  better  to  buy  a 
few  good  instruments  without  a case  than  spend  the  same 
money  in  both  case  and  instruments,  as  the  added  cost  of 
the  former  must  make  the  latter  inferior  than  they  other- 
wise would  be.  In  many  circumstances,  as  at  large  schools, 
some  means  of  putting  instruments  away  and  keeping  those 
of  various  owners  separate  is  essential,  and  some  form  or 
other  of  case  largely  adds  to  their  preservation  and  porta- 
bility. Where  a wooden  case  is  too  expensive  or  too  bulky, 
it  is  often  a good  plan  to  wrap  the  instruments  carefully  in  a 
piece  of  wash  leather,  but  pocket  cases  of  thin  wood,  covered 
with  leather,  and  having  their  corners  rounded,  are  inexpen- 
sive and  very  useful,  as  they  contain  only  the  most  essential 
instruments,  and  in  the  most  portable  form. 

252.  The  box  most  ordinarily  used  in  schools  and  by  be- 


BOXES  OF  INSTRUMENTS, 


115 

ginners  generally  is  about  seven  inches  by  four,  is  made  of 
mahogany  or  rosewood,  fastened  by  a catch,  and  contains  the 
following  instruments : — Large  compass,  that  can  be  used  as  a 
pair  of  dividers,  and  having  movable  joints  for  either  pencil  or 
pen  work,  bow  pencil  and  Low  pen,  ruling  pen,  and  a six-inch 
rule,  having  various  scales  on  it,  and  available  as  a protractor. 
Such  a box  should  cost  about  fifteen  shillings  to  eighteen 
shillings  and  sixpence.  Wherever  electrum  is  used  instead 
of  brass,  or  ivory  instead  of  boxwood,  the  cost  is  considerably 
increased,  and  such  points  as  whether  the  compasses  are 
single  or  double  jointed,  good  lock  and  key  or  no  lock  at 
all,  and  many  other  details,  modify  the  price  in  all  sorts  of 
w^ays. 

253.  It  is  needless  to  go  through  all  these  details ; we  will 
therefore  suppose  that,  having  sent  off  our  schoolboy,  in  tha 
last  paragraph,  with  a suitable  box,  a friend  wishes  to  know 
what  he  is  likely  to  get  for  the  five  pounds  that  he  is  willing 
to  spend.  On  reference  to  a maker  s price  list,  we  find  that  he 
may  have  a good  walnut  and  silk-lined  case,  tumbler  lock,  and 
the  following  electrum  instruments: — Double-jointed  com- 
passes, lengthening  bar,  pen  and  pencil  points,  double-jointed 
pencil  with  bow  compasses,  good  pair  of  dividers,  a set  of  three 
spring  bows,  two  drawing  pens,  a pricker,  a knife  key  and  an 
ivory  sector,  parallel  rule,  and  protractor.  The  highest  price 
in  this  catalogue  we  note  is  thirty-five  pounds,  so  it  will 
readily  be  seen  that  there  is  abundant  choice.  It  may  be  some 
consolation  to  those  who  do  not  see  their  way  to  any  great 
expenditure  to  learn  that  really  beyond  a certain  point  the 
cost  goes  in  finish  and  mountings  that  are  refinements  beyond 
the  need  of  the  beginner.  The  more  expensive  boxes,  too, 
include  colours,  Indian-ink,  sable  brushes,  and  palette. 

254.  The  novice  should,  if  possible,  get  some  one  to  advis/ 
him  what  instruments  the  nature  of  his  work  will  mostly  re- 


ii6 


MATHEMATICAL  INSTRUMENTS, 


quire,  for  though,  many  of  the  things  are  equally  useful  for  all, 
others  have  a special  application.  We  may  safely  discard 
then  what  is  not  wanted,  and  be  the  better  able  to  procure 
the  essentials. 

255.  In  our  next  chapter  we  shall  proceed  to  consider  some 
of  the  more  useful  accessories.  We  conclude  by  giving  somo 
of  the  more  important  measures.  These  should  be  familiar  to 
all  concerned  in  mathematical  work,  as  the  time  that  is  lost 
in  hunting  up  the  information  at  the  moment  it  is  required 
might  often  be  of  some  importance.  Any  one,  for  example, 
who  could  at  once  set  off  a scale  of  miles,  furlongs,  and  chains 
would  have  a great  advantage  over  another  who  had  to  ran- 
sack his  bookshelf  for  the  necessary  information. 

30^  sq.  yards  = i sq.  perch. 

40  sq.  perches  = i rood. 

4 roods  = I acre. 

640  acres  = i sq.  mile. 

To  these  may  he  added — 

100,000  sq.  links  = 10  sq.  chains. 

10  sq.  chains  = i acre. 

272I  sq.  feet  = i rod  of  brick- 
work. 

Square  measure  is  used  for  all  mea- 
surements of  surface,  all  superficial 
areas,  anything  like  flooring,  paving, 
or  meadow  land,  that  has  both  length 
and  breadth. 

Solid  Measure. 

1728  cubic  inches  = 1 cubic  foot. 

27  cubic  feet  = i cubic  yard. 

8 cubic  yards  = i cubic  fathom. 

This  measure  is  used  for  things  that  \ 
have  length,  breadth,  and  thickness 
or  depth,  as  stone,  timber,  excavations 
in  earth,  and  the  like. 


Long  Measure. 

= I inch. 

I foot. 

I yard. 

I rod,  pole,  or  perch. 
I furlong. 

I mile. 


12  lines  = 

1 2 inches  = 

3 feet  = 

yards  = 

40  perches  = 

8 furlongs  = 

To  these  may  be  added — 

4 inches  = i hand. 

6 feet  = I fathom. 

3 miles  = I league. 

In  military  drawing,  33  inches  make 
I pace.  In  questions  in  army  papers, 
scales  to  read  paces  are  often  called 
for. 

Surveyor’s  Measure. 

I link  ==  7*92  inches. 

100  links  or 
22  yards 

80  chains  = i mile. 


= I chain. 


Imperial  Square  Measure. 
144  sq.  inches  = i sq.  foot 
9 sq.  feet  = I sq.  yard. 


FIGURING  OF  MEASUREMENTS,  117 

Most  of  the  abbreviations  nsed  are  sufficiently  clear  to  need 
no  explanation,  such  as  in.  for  inch,  yd.  for  yard,  and  m.  for 
mile ; but  the  signs  ordinarily  used  by  architects  and  engineers 
for  feet  and  inches  are  not  so  self-evident.  To  mark  these,  a 
single  dash  is  put  after  a number  when  it  stands  for  feet,  and 
two  dashes  when  inches  are  intended.  One  foot  six  inches 
would  be  represented  as 'follows — i'  6",  A single  dash  after 
a figure  also  means  minutes,  but  the  general  circumstances  of 
the  case  will  always  show  which  is  meant,  and  minutes  are 
rarely,  if  ever,  standing  alone ; they  are  always  added  after 
some  number  of  degrees.  Thirty-seven  degrees  thirty  minutes 
would  be  marked  as  37°  30'. 

256.  When  several  angles  start  from  one  point,  and  there 
is  some  little  possibility  of  confusion  arising,  arcs  of  different 
radius  join  the  various  pairs  of  lines,  and  in  a break  of  the 
arc  the  number  of  degrees  is  marked.  Tig.  5 5 is  an  illustration 
of  this. 

257.  Where  dimensions  are  marked  on  a drawing,  unless 
the  distance  is  very  short,  the  distance  from  point  to  point  is 
not  only  expressed  by  the  figures  placed  between  these  points, 
but  a dotted  line  in  addition  joins  the  two,  and  an  arrow-head 
at  each  end  marks  the  termination.  This  may  be  placed  at 
any  point  where  it  gives  the  necessary  information.  In  an 
oblong,  for  instance,  the  lines  marking  length  and  breadth 
need  not  necessarily  start  from  the  middle  of  each  side : if 
printing  comes  there,  they  may  be  above  or  below,  or  to  left 
or  to  right  of  it,  or  even  outside  the  whole  thing,  so  long  as 
they  truly  indicate  the  measurement  from  point  to  point  that 
they  join.  In  fig.  56  we  have  represented  the  various  ways 
in  which  these  measurements,  &c.,  are  shown.  In  many 
drawings  the  measurements  will  be  found  by  taking  them  by 
compass  and  then  applying  them  to  scale,  as  a great  many 


ii8 


MATHEMATICAL  INSTRUMENTS. 


dotted  lines  across  a drawing  would  tend  to  confusion ; but  in 
detail  and  working  drawings  it  is  often  a convenience  to 
have  the  actual  distances,  so  that  the  workman  into  whose 
hand  they  are  put  is  at  once  able  to  see  what  the  size  of  the 
various  parts  should  be. 


( II9  ) 


CHAPTEE  XL 

Paper — Best  sizes  to  get — Cost — Hand  or  machine-made  paper — Cart- 
ridge paper  — Sizes  made  — Mounted  papers  — Straining  paper — 
Causes  of  failure — Straining  paper  on  panelled  boards — Drawing 
pins — Care  in  re-pinning  paper  down— Moisture  injurious  to  paper 
— How  to  mount  drawings  — Mounting  boards  — Over-mounts — 
Under-mounts  — Tracing  paper — Papier  vegetal — Directions  for 
tracing — Tracing  by  leaded  paper — Tracing  by  a sheet  of  glass — 
Hand-paper. 

258.  We  propose  now  to  give  some  little  attention  to  points 
that,  though  subordinate  to  our  subject,  are  nevertheless  of 
some  importance ; for  no  chain,  it  must  be  borne  in  mind,  is 
stronger  than  its  weakest  link,  and  the  best  instruments  pro- 
curable for  money  do  not  have  a fair  chance  unless  the  paper, 
the  Indian-ink,  and  all  the  other  accessories  are  good  of  their 
kind  too. 

259.  The  paper  we  are  to  use  may  naturally  engage  our 
first  consideration.  There  are  so  many  various  sizes  and 
thicknesses  of  paper  made,  that  it  is  a somewhat  difficult  task 
to  select  any  particular  make  for  special  approval,  as  all  have 
their  merits.  The  first  great  consideration  is  the  size  of  our 
drawing,  and  we  must  next  consider  whether  we  propose  to 
be  content  with  an  outline  or  whether  we  aspire  to  a coloured 
drawing.  A rather  smooth  paper  is  the  better  if  the  work  is  to 
be  only  inked-in,  and  a rather  rougher  texture  if  we  propose  to 
colour  on  it.  The  happy  medium  is  as  usual  the  safer  course. 
The  rough  texture  used  by  some  artists  for  watercolours. 


120 


MA  THEM  A TIC  A L INSTR  UMENTS. 


thoiigli  it  takes  the  colour  well  and  makes  very  effective  work, 
is  too  coarse  for  the  necessary  drawing  that  must  first  be 
done.  The  very  smooth  paper  known  as  hot-pressed  is,  on 
the  other  hand,  almost  too  glossy. 

260.  The  larger  sized  papers  are  the  stouter  in  make  and 
ordinarily  the  rougher  in  surface,  but  those  known  as  anti- 
quarian, double  elephant,  atlas,  and  imperial,  meet  with  very 
general  acceptance.  The  same  size  of  paper  may  often  be  met 
with  of  various  textures  and  thicknesses ; thus  imperial,  for 
example,  is  made  to  weighthirty  po  unds  to  the  ream,  or  ninety, 
or  one  hundred  and  forty,  and  costs  seven,  nine,  or  fourteen 
shillings  the  quire,  according  to  its  substance.  All  these 
papers,  and  the  others  used  by  artists  and  draughtsmen,  are 
considerably  cheaper  in  proportion  when  a ream  is  purchased, 
and  many  dealers  make  a decided  difference  when  even  five 
quires  are  taken.  A shilling  per  quire  would  probably  come 
off  all  the  prices  we  have  quoted  if  the  order  amounted  to 
five  quires.  All  really  good  drawings  are  made  on  hand-made 
paper,  as  the  surface  is  much  better ; it  is  also  whiter  and 
freer  from  blemishes  of  various  kinds.  Machine-made  paper, 
or  cartridge-paper,  as  it  is  generally  called,  is  much  cheaper 
than  hand-made,  and  is  really  sufficiently  good  for  a great 
many  purposes.  It  can  also  be  had  in  much  larger  pieces 
than  the  hand-made : a piece  one  hundred  yards  long  can  be 
had  if  we  wish.  Though  this  will  be  more  than  sufficient 
probably  for  anything  the  beginner  may  attempt,  it  is  a great 
advantage  to  be  able  to,  at  all  events,  have  as  large  a piece  as 
we  want.  Cartridge-paper  is,  for  this  reason,  and  its  cheap- 
ness, much  used  for  diagrams.  The  surface  is  very  good  for 
outline  work,  and  it  takes  colour  fairly  well.  It  is  made  in 
royal,  imperial,  and  double  elephant  sizes,  and  one  side  is 
always  rougher  than  the  other.  The  smooth  side  is,  under 
most  circumstances,  the  best  to  use. 


MOUNTING  AND  STRAINING  PAPER. 


I2I 


261.  Tlie  sizes  made  by  various  makers,  even  when  called 
by  the  same  name,  vary  a little,  but  the  following  dimensions 
are  in  any  case  right  within  an  inch  or  two,  and  will  therefore 
be  a sufficient  guide.  Demy  is  20  inches  by  1 5 ; medium, 
22  by  17;  royal,  24  by  19;  super-royal,  27  by  19;  imperial, 
30  by  22;  colombier,  34  by  33;  atlas,  34  by  26;  double 
elephant,  40  by  26;  and  antiquarian,  53  by  31.  The  prices 
range  from  two  shillings  per  quire  for  the  smallest  size  to 
sixty  shillings  per  quire  for  the  largest. 

262.  Where  drawings  are  of  considerable  size  and  exposed 
to  a heavy  risk  of  wear  and  tear,  this  wearing  and  tearing  is 
prevented  by  using  mounted  paper.  Cotton,  a material  called 
union,  and  brown  holland,  are  the  substances  generally  used 
as  mounts.  The  cost  is  that  of  the  paper,  plus  the  mount, 
and  the  labour  of  mounting  it,  but  it  is  often  a wiser  economy 
to  do  one  drawing  on  mounted  paper,  than  to  have  to  repeat 
the  labour  and  make  good  the  ravages  of  the  tooth  of  time 
and  the  sharper  perils  that  arise  from  the  handling  of  careless 
workmen,  or  the  breezes  that  play  around  the  elevated  posi- 
tions where  it  may  be  necessary  to  consult  the  work. 

263.  When  a drawing  is  of  considerable  size  and  likely  to 
be  some  time  in  hand,  it  is  a good  plan  to  strain  the  paper. 
It  is  much  easier,  too,  to  colour  on  a strained  surface,  as  the 
cockling  up  of  the  paper  is  avoided.  To  strain  a sheet  of 
paper  successfully  the  following  points  should  be  observed. 
The  paper  must  be  first  thoroughly  damped  with  clean  water, 
sponged  equally  all  over  it  on  the  opposite  side  to  that  which 
we  propose  to  use.  This  damping  should  not  be  excessive, 
and  the  surface  of  the  paper  should  be  as  gently  treated  as 
practicable,  or  its  texture  may  be  damaged  sufficiently  to 
spoil  fine  inking.  The  paper  must  be  weighted  at  the  corners 
if  it  shows  signs  of  wanting  to  roll  up.  As  soon  as  the  sheet 
has  absorbed  the  water,  the  shining  appearance  will  change 


122 


MATHEMATICAL  INSTRUMENTS. 


into  a dull  and  lustreless  one,  and  the  paper  is  then  ready  for 
fastening  down.  The  paper  must  be  turned  over  and  placed 
with  its  damper  face  on  the  board,  and  a straight-edge  or  “f 
square  placed  parallel  to  one  edge  after  the  other  and  about 
half  an  inch  from  it.  This  outer  strip  is  then  turned  up 
against  the  ruler  and  rubbed  with  glue,  moistened  in  hot 
water,  or  strong  fluid  gum  specially  prepared,  the  ordinary 
gum  being  too  thin  and  weak  for  the  purpose.  The  edge  is 
then  turned  down  again  on  the  board,  and  when  all  four 
sides  have  been  thus  treated,  a hand  must  be  put  to  each 
opposite  edge,  and  the  paper  very  gently  pulled  outwards 
each  way,  the  two  long  sides  being  first  treated  in  this  man- 
ner. A smooth  knife-handle  or  pencil  should  then  be  rubbed 
well  along  each  edge,  so  as  to  ensure  the  adherence  of  the 
paper  to  the  board.  We  have  ourselves  strained  many  a 
dozen  sheets,  and  very  rarely  had  a failure ; we  have  often 
used  good  strong  paste,  but  in  this  case  both  sides  should  be 
damped,  as  the  paste  takes  longer  to  dry.  The  idea  directing 
the  operations  is  that  the  paper  being  damped  will  stretch ; 
that  while  thus  stretched  it  should  be  fastened  securely 
down,  and  that  the  fastened  edges  should  be  thoroughly  dry 
before  the  central  part  begins  to  contract.  If  all  has  gone 
well,  the  paper  will,  in  two  or  three  hours'  time,  present  a 
beautifully  level  surface. 

264.  The  following  causes  of  failure  should  be  guarded 
against.  The  paper  should  not  be  fastened  down  until  it  has 
absorbed  all  the  water,  and  is  no  longer  expanding.  Any- 
thing like  a pool  in  one  place  should  be  avoided,  as  there 
will  else,  when  dry,  be  an  unsightly  water- mark.  The  middle 
part  should  not  dry  too  rapidly ; if  it  dries  before  the  edges, 
it  will  tear  them  up;  in  warm  weather  a slight  second  damp- 
ing may  be  given  to  the  whole  sheet,  except  the  edges,  about 
half  an  hour  after  the  fastening  down.  Until  the  whole 


HINTS  FOR  STRAINING  PAPER. 


123 


tiling  is  thoroughly  dry  the  board  should  not  be  moved  from 
a horizontal  position.  Any  attempt  to  hasten  matters  by 
putting  the  board  near  a fire  generally  courts  failure.  While 
the  paper  is  straining  and  drying,  it  should  be  placed  in  a 
place  where  it  will  be  free  from  dust.  Anything  of  this  sort 
falling  on  it  may  sully  and  roughen  the  surface,  and  we 
cannot  avoid  this  by  placing  anything  over  it,  as  this  would 
hinder  the  drying ; and  if  brown  paper  or  printed  material 
like  newspaper  were  used,  they  would  themselves  probably 
stain  the  sheet  while  in  its  damp  and  receptive  state. 

265.  When  all  is  ready  for  use,  a line  should  be  drawn  all 
round,  just  clear  of  the  gummed,  glued,  or  pasted  part.  This 
line  marks  the  limit  to  which  it  is  safe  to  go  in  the  drawing, 
as  the  sheet  has  to  be  cut  off  clear  of  the  fastened  edges,  and 
any  part  of  the  drawing  that  came  on  them  would  be  sacri- 
ficed. All  outside  of  this  line  is  very  useful  for  trying  the 
ruling  pen  or  the  colours  on,  as  we  have  already  pointed  out 
when  referring  to  the  use  of  the  pen. 

266.  The  use  of  the  T square  in  straining  not  only  gives  at 
once  a strip  ready  for  the  adhesive,  but  it  also  prevents  this 
adhesive  going  farther  in  from  the  edge  than  it  should  do. 
It  is  very  provoking  when  a drawing  comes  to  be  taken  off 
the  board  to  find  that  at  one  place  on  its  edge  the  gum  or 
paste  has  spread  too  far  inwards.  This  means  that  a new  cut 
must  be  taken,  another  inch,  perhaps,  all  the  way  along  must 
come  off,  and  then,  to  make  matters  symmetrical,  an  inch 
more  must  come  off  all  the  way  from  the  opposite  edge. 

267.  Before  putting  the  damp  paper  down  at  all  for 
straining,  we  should  be  careful  to  see  that  the  board  is  quite 
clean  and  free  from  dust.  Dust  is  a great  enemy  to  the 
draughtsman ; it  gets  on  the  set-square,  and  then  the  drawing 
is  soiled ; it  gets  on  the  paper,  and  the  lines  are  broken ; it 
gets  into  the  Indian-ink,  and  the  pen  gets  clogged.  It  is  a 


124 


MATHEMATICAL  INSTRUMENTS 


nuisance  that  cannot  too  rigidly  be  guarded  against;  we 
would,  in  fact,  almost  go  so  far  as  to  class  the  duster  among 
the  necessary  instruments  of  the  mathematical  draughtsman. 

268.  When  a drawing  is  finished  and  cut  off,  the  edges  that 
remain  all  round  on  the  board  should  be  removed  before 
another  piece  of  paper  is  fastened  down.  The  board  should 
be  placed  in  a horizontal  position,  and  a little  warm  water 
trickled  all  along  these  edges  from  a sponge.  In  a quarter  of 
an  hour  or  so  they  should  come  up  readily.  If  several  strips  are 
allowed  to  accumulate  one  over  the  other,  they  hinder  the 
proper  working  of  the  square,  and  are  much  more  difficult  to 
remove.  We  have  more  than  once  seen  a student  working 
hard  with  a chisel,  damaging  at  once  his  temper  and  his 
board,  in  the  laboured  attempt  to  remove  a mass  of  old 
material  that  might  much  more  readily  have  been  dealt  with 
if  attended  to  each  time  the  board  was  used. 

269.  Panel  boards  were  at  one  time  a good  deal  used,  but 
they  appear  to  have  now  gone  a good  deal  out  of  favour; 
they  are  rather  expensive.  We  refer  to  them  now  because 
they  afforded  a very  ready  way  of  straining  sheets  of  paper 
with  the  use  of  any  adhesive.  Pig.  57  will  give  an  idea  of  the 
construction ; it  will  be  seen  that  the  board  is  in  two  parts — a 
central  panel  and  a frame  fitting  tightly  all  round  it.  To  use 
it  for  straining  we  should  place  the  damped  sheet  of  paper  on 
the  panel,  and  then  press  the  frame  on  all  round  and  hold  it 
in  position  by  two  little  catches  behind.  The  paper  must 
always  be  a little  larger  than  the  panel,  or  there  would  be  no 
grip ; but  it  will  readily  be  seen  that  we  are  tied  down  to 
one  size  of  paper.  On  an  ordinary  board  we  can  fix  either  a 
small  or  a large  piece  of  paper  at  our  option,  but  the  panel 
board  gives  us  no  such  choice ; the  size  of  the  panel  rigidly 
governs  the  size  of  the  paper. 

270.  As  some  guide  to  cost,  we  will  quote  from  a trade 


DI?A  WING  PINS, 


125 


catalogue  before  us  the  prices  of  two  or  three  boards,  the  first 
price  given  being  for  an  ordinary  clamped  board,  and  the 
second  for  the  panelled.  A quarter  imperial,  or  1 3 X 9 J inch 
board,  is  is.  3d.  or  is.  lod. ; a royal,  or  25  x 20  inch  board,  is 
4s.  or  5s. ; and  a colombier,  or  36  x 25  inch  board,  is  /s.  gd,  and 
9s.  These  prices  are,  of  course,  to  be  taken  only  as  an  ap- 
proximation ; other  makers  might  be  either  slightly  cheaper 
or  dearer  than  those  given. 

271.  Tor  small  drawings,  and  those  that  will  not  be  long  in 
hand,  drawing  pins  make  a very  good  fastening.  These  are 
made  of  various  sizes,  and  vary  in  price  from  fourpence  to  a 
shilling  a dozen.  Three  illustrations  of  them  are  given  in 
fig.  58.  The  first  and  second  have  heads  that  readily  allow 
the  T square  to  pass  over  them,  while  the  milled  edge  of  the 
third  makes  it  more  easy  to  get  out  than  the  others.  Prac- 
tically there  is  not  much  difference  in  the  using,  for  the  T 
square  rarely  goes  so  near  the  top  or  bottom  of  the  drawing 
that  the  milled  edge  of  TTo.  3 becomes  any  real  check,  while 
the  difficulty  of  getting  hold  of  the  others  is  got  over  by  in- 
serting the  knife  blade  under  them. 

272.  As  drawing  pins  when  loose  do  not  make  at  all  good 
pocket  companions  with  pencils,  keys,  &c.,  as  one  realises  on 
diving  the  fingers  suddenly  into  one’s  pockets  for  anything, 
we  may  just  indicate  in  fig.  59  the  way  that  we  have  found 
amply  efficacious  and  protective  both  of  our  own  feelings 
and  the  points  of  the  pins.  We  cut  as  broad  a ring  as  we 
find  necessary  off  a common  wine-bottle  cork ; into  this  the 
points  readily  penetrate  and  do  not  draw  out  again,  and  the 
pins  may  in  this  way  be  carried  safely  for  days  or  weeks,  and 
will  always  be  at  hand  when  wanted. 

273.  Where  the  size  of  the  board  will  allow  it,  the  papers 
sliould  not  always  be  pinned  down  about  the  same  places,  or 


126 


MATHEMATICAL  INSTRUMENTS. 


else  the  board  gets  so  full  of  punctures  there  that  presently 
the  pins  fail  to  hold. 

274.  In  buying  pins,  only  the  more  solid  makes  should  be 
taken.  Some  kinds  have  the  upper  and  lower  points  screwed 
together,  but  the  screw  is  so  weak  that  the  heads  come  off 
on  a slight  provocation.  By  taking  an  altogether  dispropor- 
tionate amount  of  trouble  we  can  sometimes  screw  them  on 
again,  but  the  repair  is  only  temporary.  As  the  points,  when 
headless,  cannot  very  well  be  extracted  from  the  board,  it  is 
best  to  at  once  hammer  them  in  flush  with  the  surface. 

275.  When  a drawing  has  been  pinned  down  and  then 
removed  from  the  board  before  it  is  finished,  great  care  must 
be  exercised  in  pinning  it  down  again,  or  the  work  when 
resumed  will  not  be  ''true”  with  that  already  done.  The 
paper  must  be  placed  loosely  on  the  board  and  the  T square 
placed  in  position.  The  paper  must  now  be  gently  moved 
about  until  one  of  its  lines  agrees  with  the  edge  of  the  T 
square.  We  may  now  fasten  it  down,  feeling  secure  that  the 
lines  yet  to  be  drawn  will  be  in  harmony  with  those  already 
there.  This  seems,  when  mentioned,  an  almost  superfluous 
caution,  but  the  idea  does  not  occur  to  every  one ; we  have 
often  seen  beginners  evidently  under  the  impression  that  they 
have  merely  to  pin  the  paper  down  and  all  will  be  right. 
The  result  does  not  justify  their  confidence. 

276.  A drawing  should  never  be  attempted  on  damp  paper. 
At  some  seasons  of  the  year  the  moisture  in  the  air  affects 
paper  a good  deal,  and  a sheet  that  is  to  all  appearance  dry 
will  steam  when  held  near  a fire.  Besides  the  difference  in 
comfort  in  working  on  a dull  sodden  sort  of  surface  and  one 
that  is  pleasantly  crisp,  the  work  is  actually  hindered  by  any 
dampness,  as  the  india-rubber  cuts  up  the  surface,  and  the  ink 
lines  have  a blurred  look  instead  of  the  sharpness  of  definition 
that  is  so  desirable. 


!HE  LISWRY 
OF  THE 

iassn  c?  !Ll:*:3!S 


ON  MOUNTING  DR  A WINGS. 


127 


277.  Drawings  sometimes  require  to  be  mounted,  either 
for  exhibition  purposes  or  to  preserve  them.  Cardboard 
mounts,  or  linen  or  canvas,  are  the  materials  generally  used. 
Eeally  good  drawings  may  be  mounted  so  well  and  so  cheaply 
by  those  whose  business  it  is,  that  we  should  strongly  advise 
that  anything  of  value  should  be  handed  over  to  them,  but 
many  things  of  less  importance  are  nevertheless  the  better 
for  being  on  a mount,  and  a few  directions  will  be  useful. 

278.  In  mounting  a drawing  of  any  considerable  size,  a good 
stiff  sheet  of  cardboard  should  be  used,  what  is  called  four- 
sheet  or  six-sheet  being  the  best.  These  mounting  boards 
are  generally  sold  in  two  qualities,  best  and  seconds.  The 
latter  have  some  slight  blemish  or  speck,  but  practically  the 
drawing  when  it  is  stuck  down  often  covers  this.  The  differ- 
ence in  cost  is  very  considerable.  A dozen  four-sheet  mounts, 
imperial  size,  first  quality,  will  cost  about  9s.  gd ; the  same 
thing  in  second  quality  will  cost  5 s.  qd.  only,  and  it  takes  a 
keen  eye  to  see  any  difference  in  them  except  in  price. 

279.  Having  got  our  mount,  we  now  proceed  to  trim  our 
drawing,  if  it  is  to  be  over-mounted,  or  to  cut  the  necessary 
opening  in  the  mount  itself  if  the  work  is  to  be  under- 
mounted. In  the  first  case,  the  drawing  is  stuck  on  the 
mount  as  in  an  ordinary  carte  de  visite  as  we  receive  it  from 
the  photographers ; in  the  second  case,  it  is  behind  the  mount, 
and  shows  through  the  opening  cut  in  the  card  in  the  same 
way  that  the  carte  does  when  we  put  it  in  the  opening  pre- 
pared for  its  reception  in  our  album. 

280.  The  front  or  over-mount  is  the  easier  to  do,  but  the 
less  effective  when  done.  The  mode  of  procedure  is  as 
follows : — Having  cut  the  edges  of  our  drawing  neatly,  we 
place  it  either  by  eye  or  compasses  in  the  centre  of  the 
mounting-board,  and  then  make  a slight  pencil  mark  near  the 
top  corners.  The  drawing  is  now  placed  face  downwards  on 


128 


MA  THEMA  TICAL  INSTRUMENTS. 


a sheet  of  paper,  and  pasted  well  over.  It  will  probably  at 
once  begin  to  curl  and  cockle  up,  but  a slight  weight  here  and 
there  will  prevent  any  harm  being  done.  A sufficient  time 
must  be  allowed  to  elapse  to  make  sure  that  the  drawing  has 
stretched  as  far  as  it  will;  thick  paper  will  take  consider- 
ably longer  than  thin.  When  the  paper  is  thoroughly  limp, 
it  must  be  gently  taken  up  and  its  two  top  corners  placed  to 
the  marks  on  the  mounting-board,  and  then  the  whole  with 
a soft  cloth  pressed  down  to  the  mount.  If  the  edges  are 
pressed  tightly  first,  large  air-bladders  will  remain  in  the 
middle ; the  pressing,  therefore,  should  begin  in  the  middle, 
and  a series  of  gentle  dabbings  gradually  working  outwards 
will  force  the  air  before  them.  Even  then  the  surface  may 
not  look  quite  even,  but  all  probably  will  come  right  as  the 
whole  dries.  Gum  is  not  a good  material  to  use,  as  any  that 
is  squeezed  out  at  the  margins  has  an  unpleasant  shine  that 
betrays  it. 

281.  In  under-mounting  we  cut  an  opening  of  the  neces- 
sary size  in  the  centre  of  the  mount,  place  the  drawing  on 
the  table,  and  the  cut  mount  loosely  on  it.  We  now  shift 
the  mount  about  until  we  get  the  lines  of  the  drawing  true 
with  its  edges,  and  then  with  a certain  dexterous  knack,  that 
comes  by  practice,  we  slip  a hand  under  each  side,  and  lift  up 
both  mount  and  drawing  without  shifting  them  on  each  other, 
and  turn  them  upside  down.  As  there  may  have  been,  after 
all,  a slight  movement,  it  is  a very  good  plan  to  take  either  a 
couple  of  wafers,  or,  better  still,  an  inch  or  so  of  the  ready 
gummed  paper  that  we  find  round  the  edge  of  a sheet  of 
postage  stamps,  and  fasten  the  drawing  and  mount  roughly 
together  in  two  or  three  points.  We  are  now  enabled  to 
turn  our  work  over  again  the  proper  side  upwards,  and  note 
if  the  turning  upside  down  had  really  caused  any  shifting  of 
the  drawing  on  the  mount.  If  it  has,  we  must  simply  try 


TRA  CING-PAPER. 


129 


again;  but  if  not,  the  two  or  three  temporary  points  of 
attachment  can  remain,  as  they  will  be  covered  over  by  the 
long  narrow  strip  that  we  now  fasten  each  edge  of  the  draw- 
ing down  with. 

282.  Drawings  are  frequently  copied  or  multiplied  by 
means  of  various  methods  of  tracing.  Tracing-paper  is  suffi- 
ciently thin  and  transparent  to  allow  the  lines  of  a drawing 
placed  beneath  it  to  show  through ; where  the  drawing  will 
be  exposed  to  rough  handling,  durability  is  better  attained  by 
using  tracing-cloth.  Either  of  these  articles  are  very  cheap 
and  very  readily  procurable.  The  French  tracing-paper,  or 
papier  vegetal,  is  rendered  transparent  in  the  process  of  manu- 
facture, and  not,  as  in  the  ordinary  kind,  by  an  after  prepara- 
tion of  oil.  It  has  a greater  transparency,  a surface  that  will 
take  colour  well,  and  as  no  oil  enters  into  its  manufacture, 
it  has  neither  the  rank  smell  that  is  so  disagreeable  to  some 
persons  in  the  ordinary  make,  nor  can  there  be  any  possible 
injury  to  drawings  on  which  it  is  placed. 

283.  The  tracing-paper  should  be  held  securely  either  by 
weights  or  pins  while  a drawing  is  in  process  of  reproduction, 
or  a slight  and  undetected  movement  of  one  or  the  other  may 
throw  all  the  work  out.  As  it  is  often  difficult  to  see  how 
the  work  is  going  on,  and  whether  all  the  lines  of  the  original 
have  been  gone  over,  it  is  a considerable  assistance  to  slip  a 
sheet  of  white  paper  carefully  between  the  drawing  and  the 
tracing-paper,  when  all  the  lines  on  the  latter  become  at 
once  clearly  visible,  and  any  omission  readily  noted. 

284.  When  we  for  any  reason  remove  a piece  of  tracing- 
paper  temporarily,  and  then  wish  to  replace  it  accurately,  it 
is  a good  plan  to  draw  a line  or  two,  in  the  way  we  have 
indicated  in  fig.  60,  on  both  tracing  and  original  before  it  is 
removed.  If  the  lines  on  the  tracing-paper  and  the  drawing 
“read  into’’  each  other, the  old  position  is  restored.  In  one 

I 


130 


MATHEMATICAL  INSTRUMENTS. 


of  our  pieces  of  paper  in  the  illustration  we  see  it  correctly 
replaced,  and  in  the  other  incorrectly.  It  may  be  said,  Why 
take  all  this  trouble  when  the  lines  on  the  work  itself  are 
there  as  a guide  ? But  in  a complex  outline-drawing  it  is  often 
some  little  trouble  to  get  matters  straight  if  once  moved,  and 
in  a shaded  drawing  it  is  very  difficult  sometimes  to  see 
them  at  all. 

285.  Drawings  may  be  copied  by  first  placing  a sheet  of 
clean  paper  down,  then  on  this  another  sheet  rubbed  over 
with  black  lead,  the  leaded  side  being  turned  to  the  first 
paper,  and  then  over  all  the  drawing  to  be  transferred.  All 
these  should  be  securely  pinned  down  together,  and  then,  by 
means  of  a lead  pencil  or  the  tracer — a blunt  metal  or  agate 
point  fixed  in  a handle,  and  supplied  in  some  boxes  of  instru- 
ments— all  the  lines  of  the  original  are  gone  over  with  a suffi- 
cient pressure  to  transfer  them,  by  means  of  the  black  lead, 
on  to  the  clean  paper  beneath.  Two  or  three  copies  can  be 
made  at  one  operation  by  arranged  clean  and  black-leaded 
papers  alternately  under  the  drawing  to  be  copied.  Bor  a 
small  piece  of  blackened  paper,  a rubbing  over  with  a soft 
pencil  is  sufficient : but  for  larger  pieces  it  is  a saving  of  time 
to  avail  oneself  of  the  ordinary  black  lead  that  all  households 
contain  squares  of  for  domestic  use. 

286.  A large  sheet  of  glass  is  another  admirable  means  of 
copying.  Any  one  may  readily  test  this  by  placing  a piece  of 
clean  paper  on  a drawing,  and  then  holding  both  against  the 
window-pane. 

287.  We  cannot  close  our  remarks  on  paper  without  briefly 
referring  to  one  homely  member  of  the  family — the  waste 
piece  on  which  we  put,  or  should  put,  our  hands.  There  is 
alvrays  a certain  degree  of  exhalation  from  the  pores  of  the 
skin,  and  a piece  of  hand  paper,  regularly  used,  will  often 
save  the  paper  from  growing  unworkable  when  we  come  to 


USE  OF  BAND  PAPER. 


ink  or  colour.  This  hand  paper  should  he  often  changed ; a 
black  and  grimy  piece,  while  it  testifies  to  how  much  the 
drawing  has  escaped,  does  not  in  the  same  degree  bear  witness 
to  the  cleanliness  of  the  person  using  it.  The  hand  paper 
should  not  be  used  to  try  ink  or  colours  on,  as  there  is  a 
great  risk  that  they  may  presently  be  transferred  to  the 
drawing  from  it. 


( 132  ) 


CHAPTEE  XII. 

Pencils — Best  kinds  to  nse — How  pencils  should  be  cut — Putting  the 
pencil  in  the  mouth — Knife  and  file — Pocket  pencil — India-rubber 
— Care  in  pencilling — “Scrolling  out”  lines — Bottle-rubber — Vul- 
canised rubber  — Ink-eraser — India-rubber  never  to  be  held  in 
hand  when  not  in  use — How  to  cut  India-rubber — Knife  erasures 
to  be  avoided — Stale  bread — Indian-ink — How  to  select  it — How 
to  prepare  it  for  use — Liquid  ink — Nests  of  saucers — Common  ink 
to  be  avoided — Common  pen  and  crowquill — Printing — How  to 
space  out  lettering — Useful  alphabets — Lettering  square — Arbitrary 
signs  in  topographical  work — All  drawings  to  be  signed  and  dated 
— Stencilling — Colouring — Ox-gall — Recognised  colours  for  various 
materials  — Prepared  liquid  colours  — Brushes  required — Great 
cleanliness  necessary — How  to  choose  a brush — Camel  hairs — 
Sables — Sizes  and  prices — Closing  remarks. 

288.  The  pencils  that  are  used  in  mathematical  drawing 
call  for  our  consideration,  as  they  affect  the  quality  of  the 
work.  If  the  pencil  has  too  soft  a character,  it  makes  lines 
that  are  deficient  of  the  necessary  clearness,  and  the  surplus 
black  lead  rubs  and  soils  the  drawing.  It  is  difficult,  too,  to 
ink-in  satisfactorily  when  the  pencilling  is  strong  and  heavy, 
for  then  the  lines  are  already  almost  as  dark  as  ink  lines 
would  be.  When,  on  the  other  hand,  too  hard  a pencil  is 
used,  it  makes  a groove  in  the  paper ; and  this,  again,  is  fatal 
to  good  inking.  For  ordinary  work  a pencil  of  moderate 
hardness  should  be  taken,  either  an  F or  H,  or  HH.  Besides 
the  greater  clearness  and  lightness  of  the  lines  that  such 


HOW  PENCILS  SHOULD  BE  CUT. 


133 


pencils  would  make,  they  also  keep  their  points  much  better 
and  for  a longer  time  than  the  softer  kinds ; this  is  itself  a 
great  advantage,  as  it  is  a heavy  drawback  to  be  obliged  to 
stop  often  to  renew  the  points. 

289.  The  pencils  of  one  particular  maker  should  be  adhered 
to  if  they  give  satisfaction,  as  the  standard  of  hardness  or  the 
reverse  varies  with  different  men.  The  HB  of  Eowney  is 
about  equal  to  the  F of  Gilbert,  and  the  HB  of  the  latter  is 
about  equivalent  to  the  B of  the  former,  and  the  same  varia- 
tion may  be  noticed  with  other  makers.  Each  has  a con- 
sistently graduated  scale  of  his  own,  but  the  similar  letters  of 
different  men  do  not  represent  identical  values.  As  the 
colours  of  the  wood  vary  as  well,  a further  difficulty  arises,  as 
we  soon  find  when  we  pick  up  a pencil  that  from  its  colour 
we  take  to  be  an  H,  and  then  find  it  to  be  the  B or  BB  of 
some  other  maker. 

290.  The  way  the  pencil  is  cut  is  a point  not  beneath  our 
notice.  Many  draughtsmen  prefer  what  is  called  a chisel 
point,  as  they  allege  that  it  keeps  longer  in  serviceable  order 
without  attention  than  any  other.  It  is  represented  in  the 
first  illustration  in  fig.  61.  Our  view  is  a front  one.  Had 
we  shown  a side  view  of  it,  the  appearance  would  be  very 
like  that  of  the  centre  figure ; it  is,  in  fact,  very  like  the  end 
of  a chisel,  broad  one  way  and  narrow  the  other.  We  cannot 
ourselves  say  that  we  prefer  this  form.  It  requires  holding 
always  in  one  position,  and  it  is  not  well  adapted  for  marking 
a point.  On  the  whole,  we  think  that  the  central  figure 
represents  the  most  useful  form,  the  wood  cut  equally  all 
round,  and  the  lead  brought  to  the  form  of  an  acutely  pointed 
cone.  The  wood  should  be  cut  some  distance  up,  so  that  the 
view  of  the  point  is  not  obstructed  when  the  pencil  is  applied 
to  a ruler.  The  third  figure  represents  what  to  avoid.  Instead 
of  the  gently  tapering  form  of  the  central  figure,  the  cedar  is 


134 


MATHEMATICAL  INSTRUMENTS, 


gashed  irregularly,  and  the  lead,  being  for  some  distance 
without  the  support  of  the  wood,  breaks  off  directly  any 
pressure  is  laid  on  it. 

291.  Some  beginners  have  a great  fancy  for  biting  and 
chewing  the  uncut  end  of  the  pencil ; the  habit  can  scarcely 
be  defended,  though  perhaps  one  could  scarcely  point  out  that 
it  was  actually  injurious  to  work ; but  the  habit  of  putting 
the  lead  end  in  the  mouth  is  distinctly  objectionable.  The 
theory  is  that  it  enables  one  to  make  blacker  marks,  but  if 
the  pencil  in  use  does  not  give  sufficient  strength,  it  should 
be  changed  for  one  that  will. 

292.  A combination  of  knife  and  file  is  often  placed  in  the 
instrument  box,  and  either  alone  would  be  very  useful.  A 
sharp  knife  is  a very  valuable  auxiliary.;  many  a pencil  is  cut 
half  away  for  want  of  it,  as  a dull  blade  needs  undue  pressure, 
and  the  lead  under  these  circumstances  keeps  breaking  off. 
When  a good  point  has  once  been  made,  a piece  of  sandpaper 
or  a fine  file  will  be  found  very  useful,  as  a gentle  rubbing  on 
either  of  these  keeps  it  in  good  workable  condition.  Our 
objection  to  the  combination  of  the  two  arises  from  the  fact 
that  one  can  scarcely  with  comfort  use  more  than  one  of 
them.  If  we  chose  to  ignore  the  file  and  consider  it  merely 
as  the  handle  of  the  knife,  we  can  do  so ; or  if  we  choose  to 
use  the  file  and  let  the  knife  alone,  we  have  reason  equally  on 
our  side ; but  in  a thing  necessitating  so  much  cleanliness  as 
drawing,  it  hardly  seems  well  to  rub  the  file  well  over  with 
lead  and  then  grasp  it  as  a knife-handle. 

293.  There  is  much  more  command  over  a long  pencil  than 
a short  one.  When  it  has  got  cut  down  to  about  three  inches 
long,  it  should  be  put  on  the  retired  list  or  fitted  in  a porte- 
crayon.  These  short  bits  are  very  useful  as  pocket-pencils 
for  jotting  down  memoranda,  or  they  may  be  pared  down  for 
use  in  the  compasses. 


ON  RUBBING  OUT  LINES, 


135 


294.  The  owner  of  a convenient  little  piece  of  pocket- 
pencil  may  well  he  excused  if  he  shows  a slight  reluctance  to 
lend  it.  It  is  not  the  intrinsic  value  of  it,  for  on  that  ground 
the  quarter  of  a pencil  that  cost  twopence  when  new  can 
hardly  be  highly  prized;  but  this  very  valuelessness  of  it 
makes  the  borrower  careless  about  returning  it.  It  is  put 
down  and  forgotten,  and  an  hour  afterwards,  when  we  are 
far,  perhaps,  from  our  base  of  supplies,  we  feel  a pressing 
need,  and  lose  a valuable  memorandum  for  the  want  of  it. 

295.  Pencils  naturally  in  turn  suggest  india-rubber.  Where 
a drawing  has  to  be  coloured  it  should  undergo  as  little 
rubbing  as  possible,  and  a little  judgment  in  the  pencilling-in 
will  often  save  a great  deal  of  this.  Lines  should  not  be 
recklessly  carried  on  far  beyond  their  true  terminations,  and 
very  often  a mere  mark  at  the  commencement  of  a line  will 
be  sufficient  guide.  In  drawing  the  plan  of  a staircase,  for 
instance,  we  may,  when  we  have  marked  in  the  lines  of  the 
sides  and  of  the  top  and  bottom  steps,  merely  make  a pencil 
tick  at  the  right  places  for  all  the  other  steps,  and  then  at 
once  draw  them  in  ink.  Any  error  made  in  the  inking  is 
fatal,  so  that  any  inking-in  without  preliminary  pencilling 
requires  to  be  done  very  discreetly. 

^ 296.  The  need  for  the  india-rubber  becomes  much  less  when 

' care  has  been  taken  all  along  to  keep  the  drawing  as  clean  as 
possible.  The  use  of  hand-paper,  and  of  pieces  pinned  over 
any  part  not  being  worked  on,  will  greatly  lessen  the  labour 
of  cleaning  the  drawing  up. 

o 297.  When  a slight  mistake  is  made  in  the  pencilling-in,  it 
is  better  just  to  ''  scroll  it  out,’’  as  it  is  termed — i.e.,  run  a 
waved  pencil  line  along  it  to  show  that  it  is  an  error — than  to 
use  the  india-rubber.  This  latter  will  often  obliterate  more 
than  is  wanted. 

298.  Native  or  bottle  rubber  is  the  best  to  use,  as  it  does 


136 


MA  THEM  A TICAL  INSTRUMENTS. 


not  disturb  the  surface  of  the  paper  so  much  as  any  other,  but 
vulcanised  rubber  may  be  employed  where  no  tinting  has  to 
follow.  The  ink-eraser  ordinarily  cuts  up  the  paper  too  much 
to  make  one  willing  to  commend  itl  With  ordinary  care  no 
line  need  be  drawn  that  native  rubber  will  not  remove  if 
necessary.  This  native  rubber  is  sold  by  weight ; the  price 
fluctuates  a good  deal.  It  is  cut  up  into  convenient  pieces 
and  sold  under  the  name  of  the  number;  thus  6o  rubber 
means  that  one  gets  that  number  in  a pound  weight.  In  40 
rubber  the  pieces  are  of  course  larger,  as  there  are  fewer  of 
them  to  make  up  the  weight. 

299.  That  which  has  to  clean  the  drawing  should  itself  be 
clean.  It  should  not  be  allowed  to  stand  on  a table  and 
gather  dust ; above  all  things,  it  should  not  be  held  in  the 
hand,  as  the  moisture  of  the  skin  affects  it  very  injuriously. 
In  cold  weather  it  is  often  over  hard  and  rigid  for  comfort- 
able use,  but  ten  minutes  in  the  waistcoat  pocket  will  remedy 
this. 

300.  Wlien  a piece  of  india-rubber  has  got  so  dirty  at  the 
edges  that  it  soils  rather  than  cleans  the  paper,  it  need  not 
too  hastily  be  discarded,  as  all  that  is  required  is  a new 
rubbing  surface.  A slight  slicing  all  round  will  give  us  as 
good  as  a new  piece.  It  is  difficult  to  cut,  but  if  the  blade  of 
the  knife  be  flrst  dipped  in  water  it  will  go  through  much 
more  readily.  Care  must  be  taken  to  see  that  the  india- 
rubber  is  dry  before  it  is  applied  to  the  drawing. 

301.  Wrong  lines  should  not,  if  possible,  be  taken  out  with 
the  knife.  Even  a too  lavish  use  of  the  rubber  betrays  itself 
when  colour  is  applied,  and  the  action  of  the  knife,  whether 
we  ink  or  colour  over  the  place  where  it  has  been  used,  is 
ordinarily  still  more  conspicuous. 

302.  Stale  bread  makes  an  excellent  means  of  cleaning  up 
a drawing.  It  will  hardly  remove  pencil  lines  of  any  strength, 


PREPARING  INDIAN-INK  FOR  USE, 


137 


but  it  clears  up  the  general  surface  of  the  paper  beautifully. 
Crust  should  be  avoided,  as  it  may  dent  and  scratch  the 
paper  when  rubbed  on  it,  and  great  care  should  be  exercised 
that  no  particle  of  buttei  or  anything  of  that  sort  should  he 
on  the  bread.  The  knife  that  cuts  it  must  likewise  be  above 
suspicion.  The  crumbs  make  rather  a mess ; they  should  be 
carefully  collected,  or  they  get  under  the  paper,  into  the  ink 
and  in  other  ways  prove  a nuisance. 

303.  Good  ruling -pens  and  compasses  are  of  little  use  un- 
less the  ink  proves  itself  good  too.  The  best  ink  should, 
when  rubbed,  have  a soft  pasty  feel ; draughtsmen  often  try 
it,  when  choosing  a piece,  by  rubbing  it  on  their  thumb-nail, 
and  if  it  gives  a granular  sensation  it  is  at  once  discarded. 
It  ought  not  to  settle  at  all  in  water.  Though  ordinarily 
called  Indian-ink,  we  draw  our  supplies  from  China.  A good 
deal  is,  no  doubt,  made  nearer  home  than  this,  but  the  best  is 
that  of  the  Celestial  Empire.  In  ordering  a good  large  quan- 
tity we  often  get  it  in  the  original  boxes. 

304.  In  preparing  ink  for  use  one  or  two  points  deserve  our 
consideration.  A small  quantity  of  water,  three  or  four  drops, 
should  be  first  dropped  on  the  slab,  and  then  the  ink  gently 
ground;  more  water  can  be  added  by  degrees.  Warm  water 
is  better  than  cold,  as  the  ink  rubs  more  readily.  The  sides 
of  the  stick  of  ink  should  be  as  little  wet  as  possible,  or  they 
will  chip  into  the  slab ; for  this  reason  the  ink  should  be 
rubbed  into  water  that  has  been  put  into  the  slab  by  a 
brush ; the  ink  itself  should  not  supply  the  needful  supply 
of  moisture  by  being  dipped  into  the  cup  or  glass.  This 
dipping  wets  far  more  of  the  ink  than  is  at  all  desirable,  and 
as  it  afterwards  drys  it  cracks.  After  a rubbing  the  ink 
should  not  be  put  on  the  table  even  temporarily,  or  some  set- 
square  or  ruler  will  presently  bear  testimony  on  the  drawing 
to  the  fact.  It  should  at  once  be  dried  and  put  away. 


138 


MA  THEM  A TICAL  INSTRUMENTS. 


305.  Indian-ink  is  an  exceedingly  quick  dryer;  almost  as 
soon  as  a line  has  been  ruled  the  T square  may  be  fearlessly 
passed  over  it.  This  is  a great  advantage,  as  no  time  is  lost 
in  waiting.  Various  preparations  of  liquid  ink  are  made,  but 
they  cannot  be  commended.  The  most  satisfactory  plan  is  to 
mix  the  ink  afresh  every  day,  but  ''  nests  ’’  of  saucers  are 
readily  procurable,  and  their  use  saves  a great  deal  of  trouble. 
The  form  of  them  is  somewhat  like  that  of  the  common  saucer, 
but  they  have  a much  thicker  rim,  and  one  can  be  placed  on 
another,  or  half-a-dozen  on  each  other,  if  need  be.  The  junc- 
tion is  air-tight,  and  any  ink  or  colour  that  is  covered  over 
neither  evaporates  nor  contracts  dust.  Ink  rubbed  in  one  of 
these  and  covered  over  will  keep  in  good  working  condition 
for  a long  time.  These  saucers  are  sold  in  a nest  or  set  of 
six,  the  cost  of  such  a set  being  about  two  shillings.  If 
either  ink  or  colours  at  any  time  refuse  to  work  kindly  owing 
to  greasiness  of  the  paper,  a small  quantity  of  ox-gall  mixed 
with  them  will  at  once  remedy  matters.  A preparation  of 
this  can  readily  be  procured  at  any  artists’  colourman’s. 

306.  Common  ink  should  not  be  used  with  mathematical 
instruments.  It  cannot  be  tinted  over,  it  takes  a long  time 
to  dry,  and,  worst  of  all,  it  corrodes  and  deteriorates  the  nibs. 
It  should  never  appear  on  any  drawing  worked  in  Indian- 
ink  : the  common  ink  has  a bluish  tinge  in  its  blackness,  the 
other  is  distinctly  brownish,  and  the  one  will  always  look 
badly  when  placed  with  the  other.  No  measurements  should 
be  written  in  with  it,  no  printing  done  with  it. 

307.  A common  pen  is  often  necessary  to  draw  in  little 
details,  to  figure  the  dimensions,  do  printing  with,  and  so  on. 
What  is  termed  a crowquill  is  sometimes  recommended,  but 
they  are  very  weak  and  unsatisfactory  things,  and  an  ordi- 
nary fine-nibbed  pen,  in  an  ordinary  holder  that  one  can 
handle  firmly  and  comfortably,  is  far  preferable. 


ON  SETTING  OUT  PRINTING, 


139 


308.  Printing  is  a thing  that  the  beginner  will  do  well  to 
practise  a good  deal,  for  many  a good  drawing  is  spoilt  by 
bad  printing.  The  forms  of  lettering  are  not  very  numerous, 
but  in  using  them  a due  consistency  should  be  observed. 
Even  the  novice  would  hardly  print  in  the  necessary  headings, 
&c.,  to  a classic  design  in  Gothic  text.  As  a rule,  the  plainer 
the  printing  the  better. 

Where  two  or  three  lines  of  varying  length,  as 
WEST  ELEVATION, 

CHUKCH  OF  ST.  JOHN  THE  EVANGELIST,  CAREINGTON  PAEVA, 
YOEKSHIEE, 

are  put  one  under  the  other,  as  shown  above,  an  upright  line 
should  be  lightly  drawn  in  the  centre  of  the  space  the  print- 
ing is  to  occupy,  while  double  lines  are  drawn  as  guides  for 
the  tops  and  bottoms  of  the  letters.  The  more  important 
words  in  any  printing  should  receive  larger  letters,  and  a 
good  clear  space  should  always  be  left  between  the  successive 
lines. 

309.  To  set  the  words  out  with  due  regard  to  symmetry 
the  letters  in  each  line  must  be  counted,  and  a distance  equal 
to  a letter  included  for  the  space  between  each  of  the  words. 
This  number  must  then  be  halved,  and  the  necessary  number 
of  dots  marked  off  on  each  side  of  the  upright  line.  Equal 
distances  between  the  dots  should  be  allowed,  for  though 
some  letters,  like  W or  M,  are  wider  than  the  average,  others, 
like  I or  J,  take  less  space.  These  equal  distances  may  very 
conveniently  be  taken  off  by  means  of  a scale  that  reads  to 
the  edge  being  applied  to  the  line.  By  using  the  scale  we 
can  see  at  once,  by  counting  the  divisions,  how  far  the  text 
will  stretch,  and  if  the  distance  appears  too  much  or  too  little, 


i 


140  MATHEMATICAL  INSTRUMENTS. 


another  scale  can  be  tried.  This  is  better  than  marking  off  a 
number  of  divisions  by  the  eye,  and  then  finding  them  after 
all  too  far-stretching  or  too  cramped,  and  having  to  renew 
them  once  or  twice. 

310.  If  we  take  the  first  line  of  our  illustrative  example, 
we  shall  find  that  ''west  elevation”  contains  thirteen  letters, 
and  a space  between  the  words ; fourteen  divisions  then  are 
wanted,  seven  on  either  side  of  the  upright  line.  In  the 
second  line  we  find  that  there  are  forty-two  letters  and  seven 
spaces.  This  represents  forty-nine  parts.  If  we  halve  this 
number,  we  shall  find  that  there  will  be  a letter  in  the  middle, 
and  twenty-four  letters  or  spaces  on  each  side.  The  A in 
Evangelist  is  the  letter  that  comes  on  the  upright  line.  The 
name  of  the  county  contains  nine  letters,  the  central  upright 
will  therefore  pass  through  the  S.  This  general  indication  of 
the  positions  of  the  different  letters  being  obtained,  each  should 
be  then  pencilled-in.  The  divisions  will  have  given  the  re- 
quired balance  and  approximate  position ; beyond  this  they 
must  not  be  too  rigidly  adhered  to ; we  may  safely  encroach 
on  the  space  allotted  to  I,  as  the  general  average  will  bring  all 
right  as  a whole. 

3 1 1.  This  spacing  out  and  sketching  of  the  letters  may  be 
done  near  the  edge  of  a piece  of  waste  paper.  This  may  then 
be  applied  to  the  lines  ruled  on  the  drawing,  and  the  forms 
at  once  marked  off  in  their  proper  position. 

312.  A modification  of  the  set-square  has  been  devised,  and 
called  the  lettering  square.  This  gives  the  slopes  for  the 
slanting  lines  of  the  W,  A,  M,  V,  Z,  &c. ; but,  ingenious  as  it 
is,  it  would  rarely  be  much  help  to  the  beginner.  So  many 
letters  like  B,  C,  D,  G,  and  J have  curves  in  them,  that  we 
cannot  but  repeat  our  advice,  and  strongly  commend  the  per- 
sistent practice  of  hand-printing.  A or  W,  with  all  their 
liutio  carefully  ruled,  may  look  very  well ; but  if  S,  the  great 


EXAMPLES  OF  ALPHABETS. 


141 


test  letter  of  one’s  handiwork,  is  abominably  bad,  the  total 
cannot  be  considered  a success. 

313.  All  printing  upon  the  actual  drawing  should  be  left 
until  the  last  thing,  and  care  must  be  exercised,  when  the 
work  is  in  colour,  in  drawing  the  guide-lines.  They  should 
be  just  dark  enough  to  be  visible,  as  stronger  lines  would 
require  more  rubbing  to  clean  them  out,  and  this  rubbing 
often  pales  the  tints,  and  leaves  an  unsightly  whitish  streak. 

314.  It  will  probably  save  our  readers  some  little  trouble 
in  hunting  up  good  examples  of  letters  if  we  here  place  before 
them  some  of  the  varieties  that  will  be  found  most  useful. 


ABCDEFGHIJKLMN 

OPQESTUYWXYZ 

2. 

ABCDEFGHI  JKLMN 
OPQRST  UVWXYZ 


ABODE FGHIJKLMNOPQ 
R8TUVWXYZ 


142 


MATHEMATICAL  INSTRUMENTS, 


ABCD  EFG  HIJKLM 
NOPQRSTUVWXYZ 

5- 

ABCDEFGHIJKLMNO 

PQRsnruvwxYZ 


6. 


7- 

8. 


EXAMPLES  OF  ALPHABETS, 


143 


9* 

abcliefg|)t)felmnopqr 
0 tub  tojcg? 


II. 


t 


144 


MATHEMATICAL  INSTRUMENTS, 


315.  The  first  alphabet  we  have  given  is  the  form  one  is 
most  accustomed  to,  but  the  difference  in  thickness  in  different 
parts  of  each  letter  is  rather  difficult  to  deal  with  in  hand- 
work. The  second  alphabet  is  for  this  reason  preferable,  and 
it  is  also  more  distinctly  legible.  This  kind  of  lettering  is  by 
architects  and  engineers  called  block-letter.  Though  ordi- 
narily erect,  it  is  sometimes,  as  in  the  third  example,  thrown 
obliquely.  The  effect  of  this  last  is  very  good  when  it  is  well 
done,  but  beginners  will  find  No.  2 easier,  as  the  slanting 
letters  look  very  unsatisfactory  unless  all  slant  equally ; and 
this,  at  the  hands  of  the  novice,  they  are  very  likely  not  to 
do.  In  the  fourth  alphabet,  the  letters  have  a little  “ heading 
and  tailing,”  but  to  our  mind  the  bold  simplicity  of  No.  2 is 
to  be  preferred.  In  No.  5 we  get  the  same  bold  type  again, 
but  the  letters  are  much  thinner  in  line.  They  could  be 
used  in  all  but  the  most  important  parts  of  the  work, 
and  will,  perhaps,  be  found  the  most  useful  of  all  our 
examples. 

316.  While  engineers  ordinarily  choose  before  every  other 
consideration  the  most  legible  alphabets,  and  very  rightly  so, 
for  the  lettering  of  their  triumphs  of  nineteenth  century  utili- 
tarianism, the  architect,  more  embued  with  the  spirit  of  the 
past,  revives  the  beauties  of  bygone  centuries  in  his.  work,  and 
very  consistently  uses  alphabets  in  harmony  with  it.  The 
remaining  examples  are  Gothic  or  mediaeval  in  character. 
Nos.  6 and  7 would  be  most  useful  ordinarily,  but  where 
greater  emphasis  is  required.  Nos.  8 and  9 would  be  employed. 
Where  we  desire  to  make  the  lettering  bold  and  conspicuous, 
and  yet  prevent  it  from  looking  over-heavy,  the  alphabets 
numbered  10  and  ii  might  advantageously  be  selected, 
though  the  doubling  of  the  lines  adds  very  considerably  both 
to  the  labour  and  the  time  required. 

317.  A very  rich  effect  may  be  produced  by  sketching  the 


USE  OF  THE  COMMON  WRITING  PEN 


145 


letters  like  those  in  the  alphabets  10  and  ii,  and  then 
filling  in  the  light  parts  of  each  with  blue  or  gold  or 
crimson. 

3 1 8.  Many  other  alphabets  might  have  been  illustrated,  but 
those  we  have  given  are  the  kinds  likely  to  be  most  generally 
useful.  Many  professional  draughtsmen  are  as  individual  in 
their  lettering  as  in  their  work,  but  the  beginner  must  be 
content  for  a while  to  stand  upon  the  ancient  paths  and 
keep  to  the  well-beaten  track.  The  eccentricities  of  genius 
have  a charm  that  is  somehow  lacking  in  the  vagaries  ol 
ignorance. 

319.  The  common  writing-pen  must  be  used  for  all  mark- 
ing-in of  dimensions.  We  sometimes  see  the  ruling-pen 
turned  on  one  side  and  used,  but  the  result  is  generally  not 
at  all  successful.  In  land-surveying,  military  plans  of  country, 
or  any  other  kind  of  topographical  drawing,  a great  use  is 
made  of  various  arbitrary  signs,  and  these  must  ordinarily 
be  drawn  in  by  hand.  Forests,  hedgerows,  marshes,  steep 
declivities  of  ground,  and  many  other  natural  or  artificial 
features,  require  to  be  marked.  Here,  too,  practice  in  free- 
hand and  general  pen-and-ink  work  will  be  found  to  be  very 
valuable.  The  student  who  would  attempt  to  suggest  the 
idea  of  a group  of  trees  by  means  of  a ruling-pen  and  T square 
would  find  himself  somewhat  hampered. 

320.  All  drawings  should  be  signed  by  the  draughtsman, 
and  the  year,  at  least,  in  which  they  were  done  should  be 
added.  The  nature  of  the  object  represented  in  the  drawing 
would  ordinarily  be  named,  as,  for  instance,  ''villa  residence/’ 
"goods  locomotive,”  "punching  machine,”  or  "parish  of  Pre- 
shute.”  Whether  plan,  elevation,  or  section  should  also  be 
stated.  If  the  drawing  is  a reproduction,  the  source  from 
which  it  was  derived  should  be  stated. 

321.  The  use  of  the  pen  is  to  some  extent  aided  or  supcr- 

K 


146 


MA  THEM  A TIC  A L INSTR  UMENTS. 


seded  by  the  process  of  stencilling,  but  the  work  is  rougher 
in  character,  and  cannot  be  altogether  received  as  a substitute. 
Two  of  its  great  recommendations  are  that  it  ensures  uni- 
formity and  saves  time. 

322.  The  stencil  plate  is  a thin  sheet  of  brass  or  copper, 
and  in  this  the  letters  are  perforated.  Whole  words  in  com- 
mon use,  such  as  ''elevation’’  or  "section,”  are  often  employed  ; 
and  where  many  drawings  of  the  same  thing  are  required, 
or  different  details  that  all  come  under  one  general  head, 
such  as  "Great  Western  Eailway,”  or  "Engineer’s  Office, 
Metropolitan  Board  of  Works,”  the  two  advantages  we 
have  named  are  very  evident,  as  much  time  is  saved,  and 
a desirable  uniformity  amongst  the  various  drawings  is 
preserved. 

323.  To  use  the  plate,  a brush  filled  with  Indian-ink, 
or  a cheaper  preparation  made  on  purpose,  is  rubbed  care- 
fully into  the  perforations.  The  colour  must  not  be  used 
in  too  fluid  a state ; the  plate  must  be  kept  firmly  pressed 
down  to  the  surface  of  the  paper,  or  some  of  the  ink  will  get 
beneath  the  edges  of  the  letters  and  destroy  all  their  sharpness 
of  form;  and  great  care  must  be  exercised  to  see  that  the 
plate  is  very  firmly  held  in  one  position  all  through.  Any 
slipping  or  shifting  would,  if  not  noticed,  spoil  the  work.  By 
a considerable  amount  of  trouble  the  plate  can  be  replaced  in 
its  proper  position,  but  prevention  is  in  such  a case  far  better 
than  any  after  remedy. 

324.  Ornamental  borderings  and  angles,  and  various  ac- 
cessory devices  towards  the  embellishment  of  a drawing,  may 
very  well  be  stencilled.  To  draw  the  corner  designs  all 
alike  would  entail  a deal  of  trouble ; to  stencil  the  four  would 
be  a very  easy  matter.  It  is  impossible  to  quote  any  price  as 
a guide  to  the  beginner,  as  there  are  so  many  styles  of  letters 
employed,  but  the  cost  in  any  case  is  slight. 


HINTS  ON  COLOURING, 


147 


325.  As  our  student  probably  looks  forward  to  a time 
when  his  vaulting  ambition  will  o’erleap  any  mere  outline 
work,  and  when  colour  shall  give  its  added  charm,  a few 
words  on  this  point  may  not  be  amiss. 

326.  Colour  is  ordinarily  applied  to  mathematical  drawings 
in  a somewhat  conventional  way,  rather  as  suggestive  of  the 
material  than  an  attempt  to  express  its  real  tint.  The  archi- 
tect expresses  brickwork  by  a tint  much  paler  than  that  seen 
in  the  actual  thing,  and  the  engineer  gives  a much  lighter 
tint  to  his  iron  than  that  of  the  material  itself.  Cheap  chromo- 
lithographic  things  err  sadly  in  the  direction  of  overdoing  the 
colour.  They  are  showy,  and  may  perhaps  attract  the  novice 
but  too  generally  their  meretricious  qualities  are  examples  of 
what  to  avoid. 

327.  The  colours  employed  should  be  good  in  quality,  and 
only  those  should  be  used  that  admit  of  transparent  washes ; 
care  must  be  taken,  too,  that  the  hand  should  always  rest  on 
a piece  of  waste  paper,  or  the  paper  will  presently  refuse  to 
take  the  colour.  ''  Prevention  is  better  than  cure ; ’’  but  if 
the  greasiness  of  surface  makes  itself  felt,  a little  ox-gall 
added  to  the  colours  will  cause  them  to  go  smoothly  on 
the  paper. 

328.  All  colours  should  be  prepared  by  daylight.  They 
can  then  be  used  either  for  day-work  or  evening ; but  colours 
mixed  by  artificial  light  rarely  look  what  we  expect  when 
seen  in  the  daytime.  Yellow  especially  should  be  rubbed 
by  daylight;  when  it  is  prepared  by  the  yellow  light  of 
oil  or  gas,  it  does  not  look  nearly  so  strong  a tint  as  it  really 
is.  A drawing  that  looks  quite  satisfactory  by  artificial 
light  will  be  many  degrees  too  bright  in  the  white  light 
of  day. 

329.  The  tints  used  by  draughtsmen  vary  somewhat, 
according  to  individual  fancy,  but  the  following  may  be 


148 


MATHEMATICAL  INSTRUMENTS. 


considered  a sufficiently  reliable  guide.  Brickwork  in  plan 
is  ordinarily  tinted  with  crimson  lake ; and  the  same  colour, 
with  the  addition  of  a little  cadmium  or  yellow-ochre  to  take 
the  pink  look  off  it,  is  used  for  brickwork  in  elevation.  Stone 
may  be  put  in  either  with  pale  grey  or  pale  brown.  Wood- 
work in  elevation  or  plan  should  be  painted  in  yellow-ochre, 
and  in  section  the  yellow-ochre  has  the  section  lines  marked 
on  it  in  burnt  sienna.  Concrete  is  expressed  by  dark  grey 
mottled  over  irregularly  with  a stronger  tint  of  the  same 
colour.  Slates  are  marked  by  washes  of  either  purple  or 
green,  the  tint  in  either  case  being  a good  deal  diluted  with 
water.  Iron  is  indicated  by  indigo,  brass  by  gamboge,  and 
copper  by  an  orange  compounded  of  gamboge  and  crimson 
lake. 

330.  In  plans  of  estates  and  building-plots  the  colours 
used  are  those  that  will  best  suggest  the  facts : thus  grass 
is  a green  composed  of  gamboge  and  Prussian  blue,  water 
is  represented  by  the  blue  alone,  earth  by  raw  sienna,  and 
so  on. 

331.  One  of  our  best  makers  of  mathematical  things  has 
introduced  a series  of  fluid  colours  adapted  for  all  the  pur- 
poses of  the  architect  and  engineer.  Cement,  clay,  asphalte, 
concrete,  glass,  granite,  deal,  oak,  mahogany,  slate,  tiles,  gun 
metal,  lead,  zinc,  and  many  other  substances,  all  have  their 
appropriate  tint,  and  such  an  arrangement  possesses  many 
obvious  advantages.  If,  for  example,  such  a set  of  tints  came 
into  general  use,  this  uniformity  of  colouring  for  any  particular 
material  would  at  once  enable  every  one  to  see  at  a glance 
what  substance  was  intended.  All  trouble  of  mixing  tints, 
too,  would  be  avoided,  and  there  would  be  no  risk  of  the  in- 
convenience of  having  to  match  a tint.  As  all  would  be 
uniform  in  strength  and  tint,  this  particular  annoyance,  and 
it  is  a serious  one  sometimes,  could  never  occur.  Such  colours, 


HINTS  ON  THE  USE  OF  COLOURS. 


149 


to  be  really  useful,  should  always  have  the  same  strength  of 
tint,  a perfect  solubility  and  freedom  from  gritty  particles  or 
sediment  of  any  kind,  an  even  flow,  a nature  that  will  not 
injuriously  affect  any  of  the  instruments  with  which  they 
may  be  brought  in  contact,  and  a perfect  permanence  of 
colour.  The  maker  claims  that  all  these  points  are  carefully 
observed. 

332.  Where  the  ordinary  colours  are  employed,  certain 
simple  rules  should  always  be  observed.  The  beginner  will 
often,  in  making  a mixture,  rub  the  second  cake  of  colour 
into  the  tint  the  first  has  already  yielded.  In  making  a 
green,  for  example,  the  yellow  may  be  first  rubbed,  and  the 
cake  of  blue  is  then  put  into  it  and  rubbed  in  turn.  By  this 
means  a good  deal  of  the  yellow  and  resulting  green  remains 
on  the  blue,  and  makes  itself  sufficiently,  obviously,  and  dis- 
agreeably felt  when  we  take  up  the  cake  at  another  time, 
either  to  mix  a purple  by  its  aid,  or  to  get  a tint  of  pure  blue. 
The  two  colours  w^e  desire  to  blend  into  one  should  first  be 
mixed  in  different  divisions  of  the  palette  or  slab,  and  then 
worked  together  by  means  of  the  brush. 

333.  Another  failing  of  beginners  is  to  take  the  colour  off 
the  cake  by  means  of  the  brush,  instead  of  rubbing  it  on  the 
slab.  This  bad  habit  makes  a hole  in  the  middle  of  the  cake 
and  splits  it  up,  and  is  very  injurious,  moreover,  to  the  brushes. 
Even  the  mere  rubbing  on  the  slab  may  be  properly  or 
wrongly  done.  We  sometimes  see  the  cake  dipped  to  half 
its  length  in  the  cup  of  water  before  being  rubbed,  and  this 
procedure  guarantees  a very  speedy  cracking  and  crumbling 
of  the  cake.  The  orthodox  way  of  going  to  work  is  to  take  a 
very  few  drops  of  water,  either  by  means  of  the  finger  or 
brush,  put  them  on  the  slab,  and  then  proceed  to  rub  the 
colour.  After  rubbing  colours  or  the  Indian-ink  they  should 
be  put  carefully  aside,  and  the  wet  ends  tilted  up  against  a 


MATHEMATICAL  INSTRUMENTS. 


150 


pencil  or  tlie  edge  of  the  slab,  or  they  will  soil  the  table,  and 
then  the  set-square  or  scale  may  transfer  the  spot  to  the 
work. 

334.  As  some  colours  are  naturally  heavier  than  others, 
the  brush  should  be  dipped  thoroughly  into  any  tint  that  has 
been  compounded,  and  at  every  few  dippings  the  whole 
should  be  thoroughly  well  stirred,  so  as  to  prevent  any  sedi- 
ment being  deposited.  If  our  readers  will  compound  an 
orange  tint  of  vermilion  and  gamboge,  and  then  let  it  stand 
aside  for  an  hour  or  so,  they  will  see  that  it  continues  to  grow 
yellower  and  yellower  in  appearance  as  the  heavier  ingredient, 
the  vermilion,  sinks  to  the  bottom ; and  just  this  same  effect 
would  be  created  on  the  drawing  if  the  stirring  process  were 
neglected  during  its  application. 

335.  The  slab  of  colour  should  be  placed  conveniently  near 
to  the  right  hand,  and  any  colour  that  has  to  be  transferred 
from  one  side  to  the  other  should  not  be  passed  over  the 
drawing. 

336.  The  brushes  should  be  of  two  or  three  different  sizes, 
so  as  to  be  adapted  to  the  varying  nature  of  the  work.  Fine 
work  cannot  very  well  be  done  with  a coarse  brush,  and,  on 
the  other  hand,  a fine  brush  would  be  useless  for  laying  a 
large  wash  of  colour.  A man  of  experience  in  the  work  has 
such  a mastery  over  his  tools  that  he  will  often  use  one 
favourite  brush  almost  all  through  a drawing,  but  the  begin- 
ner will  certainly  find  it  an  advantage  to  hm  to  have  several 
sizes,  and  to  employ  them  with  a due  regard  to  the  conditions 
of  the  work. 

337.  When  a brush  is  not  in  use  it  should  be  carefully 
washed  and  put  aside.  It  is  a terrible  hindrance  to  good 
work,  and  very  destructive  to  the  brushes  themselves,  to 
pul  them  away  dirty,  to  reappear  harsh  and  clogged  with 
paint.  Even  when  temporarily  laid  aside  during  the  pro- 


ON  THE  CHOICE  OF  BRUSHES, 


151 

gress  of  the  work  they  should  always  be  cleansed,  or  a 
brush  laden  with  blue  or  yellow  is  suddenly  dipped  into 
the  crimson  tint,  to  the  great  damage  probably  of  both  tint 
and  temper. 

338.  In  selecting  a brush,  it  should  be  dipped  slightly  in 
water  or  moistened  between  the  lips.  If  it  forms  a good 
point,  and  maintains  its  elasticity  when  gently  pressed  on  the 
thumb-nail,  it  will  probably  prove  serviceable ; but  if  it  per- 
sists in  forming  two  or  three  separate  points,  or  keeps  any 
bend  into  which  it  is  placed,  then  the  sooner  it  is  discarded 
the  better.  Brushes  should  be  renewed  from  time  to  time,  as 
it  is  really  very  poor  economy  to  work  with  them  when  they 
grow  stumpy  and  ragged. 

339.  The  brushes  ordinarily  employed  in  all  water-colour 
work  are  either  made  of  camel-hair  or  of  sable.  The  sables 
are  of  two  kinds,  the  red  and  the  brown.  Some  authorities 
tell  us  that  the  brown  sable  brushes  are  much  the  better, 
while  others  hold  that  there  is  little  difference  really  in  their 
quality.  The  red  sables  are  much  the  cheaper,  and  our  advice 
to  our  readers  would  certainly  be  to  buy  those  until  the  dis- 
puted point  be  settled,  and,  indeed,  afterwards  too,  for  the 
red,  in  our  opinion,  answers  all  the  necessary  requirements  as 
thoroughly  as  one  could  wish.  Camel-hair  is  much  cheaper 
than  either  kind  of  sable ; it  has  not,  however,  the  wear  nor 
the  elasticity  in  it  of  the  others,  and  is  chiefly  used  in  the 
large  brushes  used  for  laying  extensive  surfaces  of  colours. 
Sable  brushes  of  the  necessary  size  for  this  work  would  be 
very  costly,  and  for  this  purpose  the  camel-hair  is  quite  good 
enough. 

340.  I'he  sizes  known  to  the  trade  as  large  swan-quill, 
goose-quill,  and  duck-quill  would  be  very  useful  sables  to 
get.  The  first  would  cost  about  5 s.  or  3s.,  according  to 
whether  we  selected  brown  sable  or  red ; the  second  would  be 


152 


MATHEMATICAL  INSTRUMENTS, 


either  about  lod.  or  /d.,  and  the  third  would  be  perhaps  6d. 
or  5d.  When  we  get  to  larger  sizes  the  prices  increase  a 
good  deal,  but  a good  camel-hair  brush  for  large  washes,  a 
flat  brush  having  the  hair  two  inches  wide,  would  only  cost 
a shilling. 

341.  Any  directions  as  to  the  actual  methods  of  tinting 
and  shading,  the  leaving  or  picking  out  of  lights,  the  manage- 
ment of  reflected  light,  stippling,  sponging,  &c.,  would,  to  be 
worth  anything  at  all,  entail  a far  larger  amount  of  space  than 
their  subordinate ' position  in  a manual  on  mathematical  in- 
struments would  justify.  We  must  content  ourselves  with 
the  broad  outline  of  the  subject,  and  refer  our  readers  to 
other  works  for  this  special  information,  or,  better  still, 
advise  them,  if  possible,  to  see  some  practised  hand  actually 
at  work. 

342.  In  the  hope  that  the  details  we  have  been  able  to 
give  and  the  points  of  practical  importance  we  have  brought 
forward  may  prove  of  use  to  the  beginner  and  save  him  much 
needless  trouble,  we  commend  our  labours  to  our  readers. 
Many  of  the  points  may  appear  trivial  and  self-evident,  but 
every  one  has  to  make  his  own  beginning,  and  facts  that  have 
long  grown  into  truisms  to  the  experienced  hand  are  per- 
plexing novelties  to  the  novice.  We  can  only  hope,  in  con- 
clusion, that  our  explanations  and  suggestions  may  prove  as 
helpful  to  the  student  as  it  has  been  our  sincere  desire  to 
make  them. 


PKINTKH  BY  BALLANTYNE,  HANSON  AND  (,'0. 
EDINBURGH  AND  LONDON 


; vJ.V 


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